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Theorem htpyi 25006
Description: A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
Hypotheses
Ref Expression
ishtpy.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
ishtpy.3 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
ishtpy.4 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
htpyi.1 (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
Assertion
Ref Expression
htpyi ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))

Proof of Theorem htpyi
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 htpyi.1 . . . 4 (𝜑𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
2 ishtpy.1 . . . . 5 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 ishtpy.3 . . . . 5 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
4 ishtpy.4 . . . . 5 (𝜑𝐺 ∈ (𝐽 Cn 𝐾))
52, 3, 4ishtpy 25004 . . . 4 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
61, 5mpbid 232 . . 3 (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
76simprd 495 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
8 oveq1 7438 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9 fveq2 6906 . . . . 5 (𝑠 = 𝐴 → (𝐹𝑠) = (𝐹𝐴))
108, 9eqeq12d 2753 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹𝑠) ↔ (𝐴𝐻0) = (𝐹𝐴)))
11 oveq1 7438 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12 fveq2 6906 . . . . 5 (𝑠 = 𝐴 → (𝐺𝑠) = (𝐺𝐴))
1311, 12eqeq12d 2753 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺𝑠) ↔ (𝐴𝐻1) = (𝐺𝐴)))
1410, 13anbi12d 632 . . 3 (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ↔ ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴))))
1514rspccva 3621 . 2 ((∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ∧ 𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
167, 15sylan 580 1 ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156  TopOnctopon 22916   Cn ccn 23232   ×t ctx 23568  IIcii 24901   Htpy chtpy 24999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235  df-htpy 25002
This theorem is referenced by:  htpycom  25008  htpyco1  25010  htpyco2  25011  htpycc  25012  phtpy01  25017  pcohtpylem  25052  txsconnlem  35245  cvmliftphtlem  35322
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