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Theorem htpyi 23884
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 23882 . . . 4 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
61, 5mpbid 235 . . 3 (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
76simprd 499 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
8 oveq1 7229 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9 fveq2 6726 . . . . 5 (𝑠 = 𝐴 → (𝐹𝑠) = (𝐹𝐴))
108, 9eqeq12d 2754 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹𝑠) ↔ (𝐴𝐻0) = (𝐹𝐴)))
11 oveq1 7229 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12 fveq2 6726 . . . . 5 (𝑠 = 𝐴 → (𝐺𝑠) = (𝐺𝐴))
1311, 12eqeq12d 2754 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺𝑠) ↔ (𝐴𝐻1) = (𝐺𝐴)))
1410, 13anbi12d 634 . . 3 (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ↔ ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴))))
1514rspccva 3543 . 2 ((∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ∧ 𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
167, 15sylan 583 1 ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  wral 3062  cfv 6389  (class class class)co 7222  0cc0 10742  1c1 10743  TopOnctopon 21820   Cn ccn 22134   ×t ctx 22470  IIcii 23785   Htpy chtpy 23877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pow 5267  ax-pr 5331  ax-un 7532
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4829  df-iun 4915  df-br 5063  df-opab 5125  df-mpt 5145  df-id 5464  df-xp 5566  df-rel 5567  df-cnv 5568  df-co 5569  df-dm 5570  df-rn 5571  df-res 5572  df-ima 5573  df-iota 6347  df-fun 6391  df-fn 6392  df-f 6393  df-fv 6397  df-ov 7225  df-oprab 7226  df-mpo 7227  df-1st 7770  df-2nd 7771  df-map 8519  df-top 21804  df-topon 21821  df-cn 22137  df-htpy 23880
This theorem is referenced by:  htpycom  23886  htpyco1  23888  htpyco2  23889  htpycc  23890  phtpy01  23895  pcohtpylem  23929  txsconnlem  32928  cvmliftphtlem  33005
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