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Theorem om1val 22638
 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 = (𝐽 ^ko 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 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om1val.o . 2 𝑂 = (𝐽 Ω1 𝑌)
2 df-om1 22614 . . . 4 Ω1 = (𝑗 ∈ Top, 𝑦 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko 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), (𝑗 ^ko II)⟩}))
4 simprl 790 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
54oveq2d 6565 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6 simprr 792 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
76eqeq2d 2620 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
86eqeq2d 2620 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
97, 8anbi12d 743 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
105, 9rabeqbidv 3168 . . . . . 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 2647 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
1413opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
154fveq2d 6107 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = (*𝑝𝐽))
16 om1val.p . . . . . . 7 (𝜑+ = (*𝑝𝐽))
1716adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → + = (*𝑝𝐽))
1815, 17eqtr4d 2647 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = + )
1918opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
204oveq1d 6564 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 ^ko II) = (𝐽 ^ko II))
21 om1val.k . . . . . . 7 (𝜑𝐾 = (𝐽 ^ko II))
2221adantr 480 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐾 = (𝐽 ^ko II))
2320, 22eqtr4d 2647 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 ^ko II) = 𝐾)
2423opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
2514, 19, 24tpeq123d 4227 . . 3 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
26 unieq 4380 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2726adantl 481 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
28 om1val.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
29 toponuni 20542 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3028, 29syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
3130adantr 480 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
3227, 31eqtr4d 2647 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
33 topontop 20541 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3428, 33syl 17 . . 3 (𝜑𝐽 ∈ Top)
35 om1val.y . . 3 (𝜑𝑌𝑋)
36 tpex 6855 . . . 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, 37ovmpt2dx 6685 . 2 (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
391, 38syl5eq 2656 1 (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173  {ctp 4129  ⟨cop 4131  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  ndxcnx 15692  Basecbs 15695  +gcplusg 15768  TopSetcts 15774  Topctop 20517  TopOnctopon 20518   Cn ccn 20838   ^ko cxko 21174  IIcii 22486  *𝑝cpco 22608   Ω1 comi 22609 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-topon 20523  df-om1 22614 This theorem is referenced by:  om1bas  22639  om1plusg  22642  om1tset  22643
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