Theorem om1val 22830

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 𝑦 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		om1val.o	. 2 ⊢ 𝑂 = (𝐽 Ω1 𝑌)
2		df-om1 22806	. . . 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 794	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5	4	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6		simprr 796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
7	6	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
8	6	eqeq2d 2632	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
9	7, 8	anbi12d 747	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
10	5, 9	rabeqbidv 3195	. . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
13	10, 12	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
14	13	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
15	4	fveq2d 6195	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = (*𝑝‘𝐽))
16		om1val.p	. . . . . . 7 ⊢ (𝜑 → + = (*𝑝‘𝐽))
17	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → + = (*𝑝‘𝐽))
18	15, 17	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = + )
19	18	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝‘𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
20	4	oveq1d 6665	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ^ko II) = (𝐽 ^ko II))
21		om1val.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (𝐽 ^ko II))
22	21	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐾 = (𝐽 ^ko II))
23	20, 22	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ^ko II) = 𝐾)
24	23	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ^ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
25	14, 19, 24	tpeq123d 4283	. . 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 4444	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
27	26	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
28		om1val.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
29		toponuni 20719	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	28, 29	syl 17	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
31	30	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
32	27, 31	eqtr4d 2659	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
33		topontop 20718	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
34	28, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
35		om1val.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
36		tpex 6957	. . . 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 6787	. 2 ⊢ (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
39	1, 38	syl5eq 2668	1 ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  {ctp 4181  ⟨cop 4183  ∪ cuni 4436  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  0cc0 9936  1c1 9937  ndxcnx 15854  Basecbs 15857  +_gcplusg 15941  TopSetcts 15947  Topctop 20698  TopOnctopon 20715   Cn ccn 21028   ^_ko cxko 21364  IIcii 22678  *_𝑝cpco 22800   Ω₁ comi 22801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-topon 20716  df-om1 22806

This theorem is referenced by:  om1bas  22831  om1plusg  22834  om1tset  22835