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Theorem om1val 22951
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 𝑦 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om1val.o . 2 𝑂 = (𝐽 Ω1 𝑌)
2 df-om1 22927 . . . 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 811 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
54oveq2d 6781 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6 simprr 813 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
76eqeq2d 2734 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
86eqeq2d 2734 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
97, 8anbi12d 749 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
105, 9rabeqbidv 3299 . . . . . 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 472 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1310, 12eqtr4d 2761 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
1413opeq2d 4516 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
154fveq2d 6308 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = (*𝑝𝐽))
16 om1val.p . . . . . . 7 (𝜑+ = (*𝑝𝐽))
1716adantr 472 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → + = (*𝑝𝐽))
1815, 17eqtr4d 2761 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = + )
1918opeq2d 4516 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
204oveq1d 6780 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 ^ko II) = (𝐽 ^ko II))
21 om1val.k . . . . . . 7 (𝜑𝐾 = (𝐽 ^ko II))
2221adantr 472 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐾 = (𝐽 ^ko II))
2320, 22eqtr4d 2761 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗 ^ko II) = 𝐾)
2423opeq2d 4516 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
2514, 19, 24tpeq123d 4390 . . 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 4552 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2726adantl 473 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
28 om1val.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
29 toponuni 20842 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3028, 29syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
3130adantr 472 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
3227, 31eqtr4d 2761 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
33 topontop 20841 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3428, 33syl 17 . . 3 (𝜑𝐽 ∈ Top)
35 om1val.y . . 3 (𝜑𝑌𝑋)
36 tpex 7074 . . . 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 6904 . 2 (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
391, 38syl5eq 2770 1 (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  {crab 3018  Vcvv 3304  {ctp 4289  cop 4291   cuni 4544  cfv 6001  (class class class)co 6765  cmpt2 6767  0cc0 10049  1c1 10050  ndxcnx 15977  Basecbs 15980  +gcplusg 16064  TopSetcts 16070  Topctop 20821  TopOnctopon 20838   Cn ccn 21151   ^ko cxko 21487  IIcii 22800  *𝑝cpco 22921   Ω1 comi 22922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-iota 5964  df-fun 6003  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-topon 20839  df-om1 22927
This theorem is referenced by:  om1bas  22952  om1plusg  22955  om1tset  22956
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