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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 32586 | . 2 ⊢ (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph‘𝐽)((0[,]1) × {(𝑓‘0)})))) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {csn 4525 class class class wbr 5030 × cxp 5517 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 [,]cicc 12729 Cn ccn 21829 IIcii 23480 ≃phcphtpc 23574 PConncpconn 32579 SConncsconn 32580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-sconn 32582 |
This theorem is referenced by: sconntop 32588 txsconn 32601 resconn 32606 iinllyconn 32614 cvmlift2lem10 32672 cvmlift3lem2 32680 cvmlift3 32688 |
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