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Theorem htpyi 25020
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 25018 . . . 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 6907 . . . . 5 (𝑠 = 𝐴 → (𝐹𝑠) = (𝐹𝐴))
108, 9eqeq12d 2751 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹𝑠) ↔ (𝐴𝐻0) = (𝐹𝐴)))
11 oveq1 7438 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12 fveq2 6907 . . . . 5 (𝑠 = 𝐴 → (𝐺𝑠) = (𝐺𝐴))
1311, 12eqeq12d 2751 . . . 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 1537  wcel 2106  wral 3059  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154  TopOnctopon 22932   Cn ccn 23248   ×t ctx 23584  IIcii 24915   Htpy chtpy 25013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-htpy 25016
This theorem is referenced by:  htpycom  25022  htpyco1  25024  htpyco2  25025  htpycc  25026  phtpy01  25031  pcohtpylem  25066  txsconnlem  35225  cvmliftphtlem  35302
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