Mathbox for Mario Carneiro |
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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 32475 | . . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | fveq1 6671 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0)) | |
3 | fveq1 6671 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1)) | |
4 | 2, 3 | eqeq12d 2839 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1))) |
5 | id 22 | . . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) | |
6 | 2 | sneqd 4581 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)}) |
7 | 6 | xpeq2d 5587 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)})) |
8 | 5, 7 | breq12d 5081 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))) |
9 | 4, 8 | imbi12d 347 | . . . 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 3622 | . . 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 507 | . 2 ⊢ (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))) |
12 | 11 | 3imp 1107 | 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 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {csn 4569 class class class wbr 5068 × cxp 5555 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 [,]cicc 12744 Cn ccn 21834 IIcii 23485 ≃phcphtpc 23575 PConncpconn 32468 SConncsconn 32469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-iota 6316 df-fv 6365 df-ov 7161 df-sconn 32471 |
This theorem is referenced by: sconnpht2 32487 sconnpi1 32488 txsconn 32490 |
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