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Theorem htpyi 23505
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 23503 . . . 4 (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))))
61, 5mpbid 233 . . 3 (𝜑 → (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠))))
76simprd 496 . 2 (𝜑 → ∀𝑠𝑋 ((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)))
8 oveq1 7152 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻0) = (𝐴𝐻0))
9 fveq2 6663 . . . . 5 (𝑠 = 𝐴 → (𝐹𝑠) = (𝐹𝐴))
108, 9eqeq12d 2834 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻0) = (𝐹𝑠) ↔ (𝐴𝐻0) = (𝐹𝐴)))
11 oveq1 7152 . . . . 5 (𝑠 = 𝐴 → (𝑠𝐻1) = (𝐴𝐻1))
12 fveq2 6663 . . . . 5 (𝑠 = 𝐴 → (𝐺𝑠) = (𝐺𝐴))
1311, 12eqeq12d 2834 . . . 4 (𝑠 = 𝐴 → ((𝑠𝐻1) = (𝐺𝑠) ↔ (𝐴𝐻1) = (𝐺𝐴)))
1410, 13anbi12d 630 . . 3 (𝑠 = 𝐴 → (((𝑠𝐻0) = (𝐹𝑠) ∧ (𝑠𝐻1) = (𝐺𝑠)) ↔ ((𝐴𝐻0) = (𝐹𝐴) ∧ (𝐴𝐻1) = (𝐺𝐴))))
1514rspccva 3619 . 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 396   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526  TopOnctopon 21446   Cn ccn 21760   ×t ctx 22096  IIcii 23410   Htpy chtpy 23498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763  df-htpy 23501
This theorem is referenced by:  htpycom  23507  htpyco1  23509  htpyco2  23510  htpycc  23511  phtpy01  23516  pcohtpylem  23550  txsconnlem  32384  cvmliftphtlem  32461
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