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Theorem htpyi 22994
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 22992 . . . 4 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
61, 5mpbid 222 . . 3 (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
76simprd 482 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
8 oveq1 6821 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9 fveq2 6353 . . . . 5 (𝑠 = 𝐴 → (𝐹𝑠) = (𝐹𝐴))
108, 9eqeq12d 2775 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹𝑠) ↔ (𝐴𝐻0) = (𝐹𝐴)))
11 oveq1 6821 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12 fveq2 6353 . . . . 5 (𝑠 = 𝐴 → (𝐺𝑠) = (𝐺𝐴))
1311, 12eqeq12d 2775 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺𝑠) ↔ (𝐴𝐻1) = (𝐺𝐴)))
1410, 13anbi12d 749 . . 3 (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ↔ ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴))))
1514rspccva 3448 . 2 ((∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ∧ 𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
167, 15sylan 489 1 ((𝜑𝐴𝑋) → ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149  TopOnctopon 20937   Cn ccn 21250   ×t ctx 21585  IIcii 22899   Htpy chtpy 22987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-top 20921  df-topon 20938  df-cn 21253  df-htpy 22990
This theorem is referenced by:  htpycom  22996  htpyco1  22998  htpyco2  22999  htpycc  23000  phtpy01  23005  pcohtpylem  23039  txsconnlem  31550  cvmliftphtlem  31627
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