Theorem om1val 25331

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 25307	. . . 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 783	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5	4	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6		simprr 785	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
7	6	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
8	6	eqeq2d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
9	7, 8	anbi12d 644	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
10	5, 9	rabeqbidv 3430	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
13	10, 12	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
14	13	opeq2d 4840	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
15	4	fveq2d 6881	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = (*𝑝‘𝐽))
16		om1val.p	. . . . . . 7 ⊢ (𝜑 → + = (*𝑝‘𝐽))
17	16	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → + = (*𝑝‘𝐽))
18	15, 17	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = + )
19	18	opeq2d 4840	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝‘𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
20	4	oveq1d 7427	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ↑ko II) = (𝐽 ↑ko II))
21		om1val.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 = (𝐽 ↑ko II))
22	21	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝐾 = (𝐽 ↑ko II))
23	20, 22	eqtr4d 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ↑ko II) = 𝐾)
24	23	opeq2d 4840	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ↑ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
25	14, 19, 24	tpeq123d 4709	. . 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 4878	. . . . 5 ⊢ (𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽)
27	26	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = ∪ 𝐽)
28		om1val.j	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
29		toponuni 23212	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
30	28, 29	syl 18	. . . . 5 ⊢ (𝜑 → 𝑋 = ∪ 𝐽)
31	30	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → 𝑋 = ∪ 𝐽)
32	27, 31	eqtr4d 2799	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
33		topontop 23211	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
34	28, 33	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
35		om1val.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
36		tpex 7751	. . . 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	ovmpodx 7563	. 2 ⊢ (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
39	1, 38	eqtrid 2808	1 ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451  {ctp 4588  ⟨cop 4590  ∪ cuni 4867  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414  0cc0 11181  1c1 11182  ndxcnx 17351  Basecbs 17367  +_gcplusg 17408  TopSetcts 17414  Topctop 23191  TopOnctopon 23208   Cn ccn 23522   ↑_ko cxko 23860  IIcii 25176  *_𝑝cpco 25301   Ω₁ comi 25302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-topon 23209  df-om1 25307

This theorem is used by:  om1bas  25332  om1plusg  25335  om1tset  25336