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

Theorem isphtpy 23512
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 21777 . . . . 5 (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
3 oveq2 7153 . . . . . . 7 (𝑗 = 𝐽 → (II Cn 𝑗) = (II Cn 𝐽))
4 oveq2 7153 . . . . . . . . 9 (𝑗 = 𝐽 → (II Htpy 𝑗) = (II Htpy 𝐽))
54oveqd 7162 . . . . . . . 8 (𝑗 = 𝐽 → (𝑓(II Htpy 𝑗)𝑔) = (𝑓(II Htpy 𝐽)𝑔))
65rabeqdv 3482 . . . . . . 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 7218 . . . . . 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 23502 . . . . . 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 7178 . . . . . . 7 (II Cn 𝐽) ∈ V
109, 9mpoex 7766 . . . . . 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 6761 . . . . 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 7154 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓(II Htpy 𝐽)𝑔) = (𝐹(II Htpy 𝐽)𝐺))
14 simpl 483 . . . . . . . . . 10 ((𝑓 = 𝐹𝑔 = 𝐺) → 𝑓 = 𝐹)
1514fveq1d 6665 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘0) = (𝐹‘0))
1615eqeq2d 2829 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((0𝑠) = (𝑓‘0) ↔ (0𝑠) = (𝐹‘0)))
1714fveq1d 6665 . . . . . . . . 9 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑓‘1) = (𝐹‘1))
1817eqeq2d 2829 . . . . . . . 8 ((𝑓 = 𝐹𝑔 = 𝐺) → ((1𝑠) = (𝑓‘1) ↔ (1𝑠) = (𝐹‘1)))
1916, 18anbi12d 630 . . . . . . 7 ((𝑓 = 𝐹𝑔 = 𝐺) → (((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2019ralbidv 3194 . . . . . 6 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝑓‘0) ∧ (1𝑠) = (𝑓‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))))
2113, 20rabeqbidv 3483 . . . . 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 482 . . . 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 7178 . . . . . 6 (𝐹(II Htpy 𝐽)𝐺) ∈ V
2524rabex 5226 . . . . 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 7291 . . 3 (𝜑 → (𝐹(PHtpy‘𝐽)𝐺) = { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))})
2827eleq2d 2895 . 2 (𝜑 → (𝐻 ∈ (𝐹(PHtpy‘𝐽)𝐺) ↔ 𝐻 ∈ { ∈ (𝐹(II Htpy 𝐽)𝐺) ∣ ∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1))}))
29 oveq 7151 . . . . . 6 ( = 𝐻 → (0𝑠) = (0𝐻𝑠))
3029eqeq1d 2820 . . . . 5 ( = 𝐻 → ((0𝑠) = (𝐹‘0) ↔ (0𝐻𝑠) = (𝐹‘0)))
31 oveq 7151 . . . . . 6 ( = 𝐻 → (1𝑠) = (1𝐻𝑠))
3231eqeq1d 2820 . . . . 5 ( = 𝐻 → ((1𝑠) = (𝐹‘1) ↔ (1𝐻𝑠) = (𝐹‘1)))
3330, 32anbi12d 630 . . . 4 ( = 𝐻 → (((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3433ralbidv 3194 . . 3 ( = 𝐻 → (∀𝑠 ∈ (0[,]1)((0𝑠) = (𝐹‘0) ∧ (1𝑠) = (𝐹‘1)) ↔ ∀𝑠 ∈ (0[,]1)((0𝐻𝑠) = (𝐹‘0) ∧ (1𝐻𝑠) = (𝐹‘1))))
3534elrab 3677 . 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, 35syl6bb 288 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cfv 6348  (class class class)co 7145  cmpo 7147  0cc0 10525  1c1 10526  [,]cicc 12729  Topctop 21429   Cn ccn 21760  IIcii 23410   Htpy chtpy 23498  PHtpycphtpy 23499
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-rep 5181  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-reu 3142  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-f1 6353  df-fo 6354  df-f1o 6355  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-phtpy 23502
This theorem is referenced by:  phtpyhtpy  23513  phtpyi  23515  isphtpyd  23517
  Copyright terms: Public domain W3C validator