MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  om1val Structured version   Visualization version   GIF version

Theorem om1val 25063
Description: The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
Hypotheses
Ref Expression
om1val.o 𝑂 = (𝐽 Ω1 𝑌)
om1val.b (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
om1val.p (𝜑+ = (*𝑝𝐽))
om1val.k (𝜑𝐾 = (𝐽ko II))
om1val.j (𝜑𝐽 ∈ (TopOn‘𝑋))
om1val.y (𝜑𝑌𝑋)
Assertion
Ref Expression
om1val (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Distinct variable groups:   𝑓,𝐽   𝜑,𝑓   𝑓,𝑌
Allowed substitution hints:   𝐵(𝑓)   + (𝑓)   𝐾(𝑓)   𝑂(𝑓)   𝑋(𝑓)

Proof of Theorem om1val
Dummy variables 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om1val.o . 2 𝑂 = (𝐽 Ω1 𝑌)
2 df-om1 25039 . . . 4 Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩})
32a1i 11 . . 3 (𝜑 → Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩}))
4 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
54oveq2d 7447 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6 simprr 773 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
76eqeq2d 2748 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
86eqeq2d 2748 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
97, 8anbi12d 632 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
105, 9rabeqbidv 3455 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
11 om1val.b . . . . . . 7 (𝜑𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1211adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1310, 12eqtr4d 2780 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
1413opeq2d 4880 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
154fveq2d 6910 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = (*𝑝𝐽))
16 om1val.p . . . . . . 7 (𝜑+ = (*𝑝𝐽))
1716adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → + = (*𝑝𝐽))
1815, 17eqtr4d 2780 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = + )
1918opeq2d 4880 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
204oveq1d 7446 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = (𝐽ko II))
21 om1val.k . . . . . . 7 (𝜑𝐾 = (𝐽ko II))
2221adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐾 = (𝐽ko II))
2320, 22eqtr4d 2780 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = 𝐾)
2423opeq2d 4880 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
2514, 19, 24tpeq123d 4748 . . 3 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗ko II)⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
26 unieq 4918 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2726adantl 481 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
28 om1val.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
29 toponuni 22920 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3028, 29syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
3130adantr 480 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
3227, 31eqtr4d 2780 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
33 topontop 22919 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3428, 33syl 17 . . 3 (𝜑𝐽 ∈ Top)
35 om1val.y . . 3 (𝜑𝑌𝑋)
36 tpex 7766 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V
3736a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V)
383, 25, 32, 34, 35, 37ovmpodx 7584 . 2 (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
391, 38eqtrid 2789 1 (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  {ctp 4630  cop 4632   cuni 4907  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156  ndxcnx 17230  Basecbs 17247  +gcplusg 17297  TopSetcts 17303  Topctop 22899  TopOnctopon 22916   Cn ccn 23232  ko cxko 23569  IIcii 24901  *𝑝cpco 25033   Ω1 comi 25034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-topon 22917  df-om1 25039
This theorem is referenced by:  om1bas  25064  om1plusg  25067  om1tset  25068
  Copyright terms: Public domain W3C validator