Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sconnpht Structured version   Visualization version   GIF version

Theorem sconnpht 32098
Description: A closed path in a simply connected space is contractible to a point. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
sconnpht ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))

Proof of Theorem sconnpht
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 issconn 32095 . . 3 (𝐽 ∈ SConn ↔ (𝐽 ∈ PConn ∧ ∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}))))
2 fveq1 6495 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘0) = (𝐹‘0))
3 fveq1 6495 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘1) = (𝐹‘1))
42, 3eqeq12d 2786 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘0) = (𝑓‘1) ↔ (𝐹‘0) = (𝐹‘1)))
5 id 22 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
62sneqd 4447 . . . . . . 7 (𝑓 = 𝐹 → {(𝑓‘0)} = {(𝐹‘0)})
76xpeq2d 5433 . . . . . 6 (𝑓 = 𝐹 → ((0[,]1) × {(𝑓‘0)}) = ((0[,]1) × {(𝐹‘0)}))
85, 7breq12d 4938 . . . . 5 (𝑓 = 𝐹 → (𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)}) ↔ 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)})))
94, 8imbi12d 337 . . . 4 (𝑓 = 𝐹 → (((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})) ↔ ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
109rspccv 3525 . . 3 (∀𝑓 ∈ (II Cn 𝐽)((𝑓‘0) = (𝑓‘1) → 𝑓( ≃ph𝐽)((0[,]1) × {(𝑓‘0)})) → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
111, 10simplbiim 497 . 2 (𝐽 ∈ SConn → (𝐹 ∈ (II Cn 𝐽) → ((𝐹‘0) = (𝐹‘1) → 𝐹( ≃ph𝐽)((0[,]1) × {(𝐹‘0)}))))
12113imp 1092 1 ((𝐽 ∈ SConn ∧ 𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = (𝐹‘1)) → 𝐹( ≃ph𝐽)((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
  Copyright terms: Public domain W3C validator