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Mirrors > Home > MPE Home > Th. List > Mathboxes > sconnpconn | Structured version Visualization version GIF version |
Description: A simply connected space is path-connected. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
sconnpconn | ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issconn 34772 | . 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∀wral 3056 {csn 4624 class class class wbr 5142 × cxp 5670 ‘cfv 6542 (class class class)co 7414 0cc0 11130 1c1 11131 [,]cicc 13351 Cn ccn 23115 IIcii 24782 ≃phcphtpc 24882 PConncpconn 34765 SConncsconn 34766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-iota 6494 df-fv 6550 df-ov 7417 df-sconn 34768 |
This theorem is referenced by: sconntop 34774 txsconn 34787 resconn 34792 iinllyconn 34800 cvmlift2lem10 34858 cvmlift3lem2 34866 cvmlift3 34874 |
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