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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sconnpht | Structured version Visualization version GIF version | ||
| Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.) |
| Ref | Expression |
|---|---|
| sconnpht | ⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issconn 35231 | . . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
| 2 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0)) | |
| 3 | fveq1 6905 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1)) | |
| 4 | 2, 3 | eqeq12d 2753 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1))) |
| 5 | id 22 | . . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) | |
| 6 | 2 | sneqd 4638 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)}) |
| 7 | 6 | xpeq2d 5715 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)})) |
| 8 | 5, 7 | breq12d 5156 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))) |
| 9 | 4, 8 | imbi12d 344 | . . . 4 ⊢ (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))) |
| 10 | 9 | rspccv 3619 | . . 3 ⊢ (∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})) → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))) |
| 11 | 1, 10 | simplbiim 504 | . 2 ⊢ (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))) |
| 12 | 11 | 3imp 1111 | 1 ⊢ ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {csn 4626 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 [,]cicc 13390 Cn ccn 23232 IIcii 24901 ≃phcphtpc 25001 PConncpconn 35224 SConncsconn 35225 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-iota 6514 df-fv 6569 df-ov 7434 df-sconn 35227 |
| This theorem is referenced by: sconnpht2 35243 sconnpi1 35244 txsconn 35246 |
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