Theorem om1val 24099

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 24075	. . . 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 767	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑗 = 𝐽)
5	4	oveq2d 7271	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (II Cn 𝑗) = (II Cn 𝐽))
6		simprr 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
7	6	eqeq2d 2749	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘0) = 𝑦 ↔ (𝑓‘0) = 𝑌))
8	6	eqeq2d 2749	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ((𝑓‘1) = 𝑦 ↔ (𝑓‘1) = 𝑌))
9	7, 8	anbi12d 630	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦) ↔ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)))
10	5, 9	rabeqbidv 3410	. . . . . 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 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)} = 𝐵)
14	13	opeq2d 4808	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(Base‘ndx), {𝑓 ∈ (II Cn 𝑗) ∣ ((𝑓‘0) = 𝑦 ∧ (𝑓‘1) = 𝑦)}⟩ = ⟨(Base‘ndx), 𝐵⟩)
15	4	fveq2d 6760	. . . . . 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 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (*𝑝‘𝑗) = + )
19	18	opeq2d 4808	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(+g‘ndx), (*𝑝‘𝑗)⟩ = ⟨(+g‘ndx), + ⟩)
20	4	oveq1d 7270	. . . . . 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 2781	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → (𝑗 ↑ko II) = 𝐾)
24	23	opeq2d 4808	. . . 4 ⊢ ((𝜑 ∧ (𝑗 = 𝐽 ∧ 𝑦 = 𝑌)) → ⟨(TopSet‘ndx), (𝑗 ↑ko II)⟩ = ⟨(TopSet‘ndx), 𝐾⟩)
25	14, 19, 24	tpeq123d 4681	. . 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 4847	. . . . 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 21971	. . . . . 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 2781	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ∪ 𝑗 = 𝑋)
33		topontop 21970	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
34	28, 33	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
35		om1val.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
36		tpex 7575	. . . 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 7402	. 2 ⊢ (𝜑 → (𝐽 Ω1 𝑌) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
39	1, 38	eqtrid 2790	1 ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  {ctp 4562  ⟨cop 4564  ∪ cuni 4836  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  0cc0 10802  1c1 10803  ndxcnx 16822  Basecbs 16840  +_gcplusg 16888  TopSetcts 16894  Topctop 21950  TopOnctopon 21967   Cn ccn 22283   ↑_ko cxko 22620  IIcii 23944  *_𝑝cpco 24069   Ω₁ comi 24070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-topon 21968  df-om1 24075

This theorem is referenced by:  om1bas  24100  om1plusg  24103  om1tset  24104