MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isphtpy Structured version   Visualization version   GIF version

Theorem isphtpy 25027
Description: Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
isphtpy.2 (𝜑𝐹 ∈ (II Cn 𝐽))
isphtpy.3 (𝜑𝐺 ∈ (II Cn 𝐽))
Assertion
Ref Expression
isphtpy (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
Distinct variable groups:   𝐹,𝑠   𝐺,𝑠   𝐻,𝑠   𝐽,𝑠   𝜑,𝑠

Proof of Theorem isphtpy
Dummy variables 𝑓 𝑔 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isphtpy.2 . . . . 5 (𝜑𝐹 ∈ (II Cn 𝐽))
2 cntop2 23265 . . . . 5 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3 oveq2 7439 . . . . . . 7 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4 oveq2 7439 . . . . . . . . 9 (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
54oveqd 7448 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
65rabeqdv 3449 . . . . . . 7 (𝑗 = 𝐽 → { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))})
73, 3, 6mpoeq123dv 7508 . . . . . 6 (𝑗 = 𝐽 → (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
8 df-phtpy 25017 . . . . . 6 PHtpy = (𝑗 ∈ Top ↦ (𝑓 ∈ (II Cn 𝑗), 𝑔 ∈ (II Cn 𝑗) ↦ { ∈ (𝑓(II Htpy 𝑗)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
9 ovex 7464 . . . . . . 7 (II Cn 𝐽) ∈ V
109, 9mpoex 8103 . . . . . 6 (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}) ∈ V
117, 8, 10fvmpt 7016 . . . . 5 (𝐽 ∈ Top → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
121, 2, 113syl 18 . . . 4 (𝜑 → (PHtpy‘𝐽) = (𝑓 ∈ (II Cn 𝐽), 𝑔 ∈ (II Cn 𝐽) ↦ { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))}))
13 oveq12 7440 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14 simpl 482 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
1514fveq1d 6909 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
1615eqeq2d 2746 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((0𝑠) = (𝑓‘0) ↔ (0𝑠) = (𝐹‘0)))
1714fveq1d 6909 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
1817eqeq2d 2746 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((1𝑠) = (𝑓‘1) ↔ (1𝑠) = (𝐹‘1)))
1916, 18anbi12d 632 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2019ralbidv 3176 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2113, 20rabeqbidv 3452 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2221adantl 481 . . . 4 ((𝜑 ∧ (𝑓 = 𝐹𝑔 = 𝐺)) → { ∈ (𝑓(II Htpy 𝐽)𝑔) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1))} = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
23 isphtpy.3 . . . 4 (𝜑𝐺 ∈ (II Cn 𝐽))
24 ovex 7464 . . . . . 6 (𝐹(II Htpy 𝐽)𝐺) ∈ V
2524rabex 5345 . . . . 5 { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V
2625a1i 11 . . . 4 (𝜑 → { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ∈ V)
2712, 22, 1, 23, 26ovmpod 7585 . . 3 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2827eleq2d 2825 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))}))
29 oveq 7437 . . . . . 6 ( = 𝐻 → (0𝑠) = (0𝐻𝑠))
3029eqeq1d 2737 . . . . 5 ( = 𝐻 → ((0𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31 oveq 7437 . . . . . 6 ( = 𝐻 → (1𝑠) = (1𝐻𝑠))
3231eqeq1d 2737 . . . . 5 ( = 𝐻 → ((1𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
3330, 32anbi12d 632 . . . 4 ( = 𝐻 → (((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3433ralbidv 3176 . . 3 ( = 𝐻 → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3534elrab 3695 . 2 (𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))} ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3628, 35bitrdi 287 1 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ (𝐻 ∈ (𝐹(II Htpy 𝐽)𝐺) ∧ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154  [,]cicc 13387  Topctop 22915   Cn ccn 23248  IIcii 24915   Htpy chtpy 25013  PHtpycphtpy 25014
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-rep 5285  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-reu 3379  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-f1 6568  df-fo 6569  df-f1o 6570  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-phtpy 25017
This theorem is referenced by:  phtpyhtpy  25028  phtpyi  25030  isphtpyd  25032
  Copyright terms: Public domain W3C validator