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Theorem om1val 23635
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 23611 . . . 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 770 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑗 = 𝐽)
54oveq2d 7151 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝑦 = 𝑌)
76eqeq2d 2809 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
86eqeq2d 2809 . . . . . . . 8 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
97, 8anbi12d 633 . . . . . . 7 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
105, 9rabeqbidv 3433 . . . . . 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 484 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
1310, 12eqtr4d 2836 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
1413opeq2d 4772 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
154fveq2d 6649 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = (*𝑝𝐽))
16 om1val.p . . . . . . 7 (𝜑+ = (*𝑝𝐽))
1716adantr 484 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → + = (*𝑝𝐽))
1815, 17eqtr4d 2836 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (*𝑝𝑗) = + )
1918opeq2d 4772 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
204oveq1d 7150 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = (𝐽ko II))
21 om1val.k . . . . . . 7 (𝜑𝐾 = (𝐽ko II))
2221adantr 484 . . . . . 6 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → 𝐾 = (𝐽ko II))
2320, 22eqtr4d 2836 . . . . 5 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → (𝑗ko II) = 𝐾)
2423opeq2d 4772 . . . 4 ((𝜑 ∧ (𝑗 = 𝐽𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
2514, 19, 24tpeq123d 4644 . . 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 4811 . . . . 5 (𝑗 = 𝐽 𝑗 = 𝐽)
2726adantl 485 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝐽)
28 om1val.j . . . . . 6 (𝜑𝐽 ∈ (TopOn‘𝑋))
29 toponuni 21519 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
3028, 29syl 17 . . . . 5 (𝜑𝑋 = 𝐽)
3130adantr 484 . . . 4 ((𝜑𝑗 = 𝐽) → 𝑋 = 𝐽)
3227, 31eqtr4d 2836 . . 3 ((𝜑𝑗 = 𝐽) → 𝑗 = 𝑋)
33 topontop 21518 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3428, 33syl 17 . . 3 (𝜑𝐽 ∈ Top)
35 om1val.y . . 3 (𝜑𝑌𝑋)
36 tpex 7450 . . . 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 7280 . 2 (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
391, 38syl5eq 2845 1 (𝜑𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {crab 3110  Vcvv 3441  {ctp 4529  cop 4531   cuni 4800  cfv 6324  (class class class)co 7135  cmpo 7137  0cc0 10526  1c1 10527  ndxcnx 16472  Basecbs 16475  +gcplusg 16557  TopSetcts 16563  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  ko cxko 22166  IIcii 23480  *𝑝cpco 23605   Ω1 comi 23606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-topon 21516  df-om1 23611
This theorem is referenced by:  om1bas  23636  om1plusg  23639  om1tset  23640
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