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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 32095 | . . 3 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | fveq1 6495 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0)) | |
3 | fveq1 6495 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1)) | |
4 | 2, 3 | eqeq12d 2786 | . . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1))) |
5 | id 22 | . . . . . 6 ⊢ (𝑓 = 𝐹 → 𝑓 = 𝐹) | |
6 | 2 | sneqd 4447 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)}) |
7 | 6 | xpeq2d 5433 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)})) |
8 | 5, 7 | breq12d 4938 | . . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)}))) |
9 | 4, 8 | imbi12d 337 | . . . 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 3525 | . . 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 497 | . 2 ⊢ (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph‘𝐽)((0[,]1) × {(𝐹‘0)})))) |
12 | 11 | 3imp 1092 | 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 1069 = wceq 1508 ∈ wcel 2051 ∀wral 3081 {csn 4435 class class class wbr 4925 × cxp 5401 ‘cfv 6185 (class class class)co 6974 0cc0 10333 1c1 10334 [,]cicc 12555 Cn ccn 21551 IIcii 23201 ≃phcphtpc 23291 PConncpconn 32088 SConncsconn 32089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-ext 2743 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-xp 5409 df-iota 6149 df-fv 6193 df-ov 6977 df-sconn 32091 |
This theorem is referenced by: sconnpht2 32107 sconnpi1 32108 txsconn 32110 |
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