proposition A.539 Universal property

open in the book · appendices/A-long-proofs.tex:26296 · p. 3055

Rests on

depends_on definition 13.49 Smooth map between manifolds ¶
- depends_on definition 13.44 Atlas ¶
  - depends_on definition 13.43 Coordinate system ¶
- depends_on definition 13.45 Differentiable map on a topological space ¶
  - depends_on definition 13.44 Atlas ¶ ↺
  - depends_on definition 13.43 Coordinate system ¶ ↺
- depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 13.47 Differentiable structure ¶
    - depends_on definition 13.44 Atlas ¶ ↺
depends_on lemma A.538 Submersions have smooth local sections ¶
- depends_on definition 13.53 Immersion, submersion, embedding ¶
  - depends_on definition 13.52 Differential; pushforward ¶
    - depends_on definition 13.49 Smooth map between manifolds ¶ ↺
    - depends_on definition 13.81 Vector on a manifold ¶
      - depends_on definition 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶ ↺
      - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 3.47 Bijective map ¶ ↺
      - depends_on definition 3.49 Composition map ¶
      - depends_on definition 3.50 Identity map ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
- depends_on definition 13.49 Smooth map between manifolds ¶ ↺
- depends_on theorem 13.63 Constant rank theorem ¶
  - depends_on corollary A.292 Inverse function theorem ¶
    - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on definition 7.13 Composite function ¶
      - depends_on definition 7.99 Differentiability at a point ¶
      - depends_on definition 7.102 Differential of a function ¶
      - proves proof ch:05-real-analysis@proof-63 ¶
    - depends_on theorem A.286 Implicit function theorem ¶
      - depends_on definition 7.99 Differentiability at a point ¶ ↺
      - depends_on proposition 7.104 Chain rule in several variables ¶ ↺
      - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
      - proves proof app:A-long-proofs@proof-185 ¶
    - proves proof app:A-long-proofs@proof-186 ¶
  - depends_on definition 7.98 Functions of class $C^{1}$ ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.97 Partial derivative; gradient ¶
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 5.15 Basis ¶
  - depends_on theorem A.286 Implicit function theorem ¶ ↺
  - proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
- proves proof app:A-long-proofs@proof-326 ¶
proves proof app:A-long-proofs@proof-327 ¶

Supports

depends_on corollary A.540 Uniqueness of the smooth structure ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Smooth map between manifolds	declared	appendices/A-long-proofs.tex:26300
`depends_on`	→	Submersions have smooth local sections	declared	appendices/A-long-proofs.tex:26300
`depends_on`	←	Uniqueness of the smooth structure	declared	appendices/A-long-proofs.tex:26318
`proves`	←	app:A-long-proofs@proof-327	declared	appendices/A-long-proofs.tex:26303