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	⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
om1val.p	⊢ (𝜑 → + = (*𝑝‘𝐽))
om1val.k	⊢ (𝜑 → 𝐾 = (𝐽 ^ko II))
om1val.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
om1val.y	⊢ (𝜑 → 𝑌 ∈ 𝑋)

Assertion

Ref	Expression
om1val	⊢ (𝜑 → 𝑂 = {⟨(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.

Step	Hyp	Ref	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), (*𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → Ω1 = (𝑗 ∈ Top, 𝑦 ∈ ∪ 𝑗 ↦ {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩}))
4		simprl 790	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5	4	oveq2d 6565	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
7	6	eqeq2d 2620	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
8	6	eqeq2d 2620	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
9	7, 8	anbi12d 743	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
10	5, 9	rabeqbidv 3168	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 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 ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
12	11	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
13	10, 12	eqtr4d 2647	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
14	13	opeq2d 4347	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
15	4	fveq2d 6107	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = (*𝑝‘𝐽))
16		om1val.p	. . . . . . 7 ⊢ (𝜑 → + = (*𝑝‘𝐽))
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → + = (*𝑝‘𝐽))
18	15, 17	eqtr4d 2647	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = + )
19	18	opeq2d 4347	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝‘𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
20	4	oveq1d 6564	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ^ko II) = (𝐽 ^ko II))
21		om1val.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (𝐽 ^ko II))
22	21	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐾 = (𝐽 ^ko II))
23	20, 22	eqtr4d 2647	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ^ko II) = 𝐾)
24	23	opeq2d 4347	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
25	14, 19, 24	tpeq123d 4227	. . 3 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → {⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩, ⟨(+g‘ndx), (*𝑝‘𝑗)⟩, ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
26		unieq 4380	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
27	26	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
28		om1val.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
29		toponuni 20542	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
31	30	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
32	27, 31	eqtr4d 2647	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
33		topontop 20541	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
34	28, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
35		om1val.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
36		tpex 6855	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V
37	36	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩} ∈ V)
38	3, 25, 32, 34, 35, 37	ovmpt2dx 6685	. 2 ⊢ (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
39	1, 38	syl5eq 2656	1 ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})

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