Theorem xpstopnlem2 22419

Description: Lemma for xpstopn 22420. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypotheses

Ref	Expression
xpstps.t	⊢ 𝑇 = (𝑅 ×s 𝑆)
xpstopn.j	⊢ 𝐽 = (TopOpen‘𝑅)
xpstopn.k	⊢ 𝐾 = (TopOpen‘𝑆)
xpstopn.o	⊢ 𝑂 = (TopOpen‘𝑇)
xpstopnlem.x	⊢ 𝑋 = (Base‘𝑅)
xpstopnlem.y	⊢ 𝑌 = (Base‘𝑆)
xpstopnlem.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})

Assertion

Ref	Expression
xpstopnlem2	⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝑇(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem xpstopnlem2

Step	Hyp	Ref	Expression
1		eqid 2821	. . . . 5 ⊢ ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
2		fvexd 6685	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V)
3		2on 8111	. . . . . 6 ⊢ 2o ∈ On
4	3	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2o ∈ On)
5		fnpr2o 16830	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o)
6		eqid 2821	. . . . 5 ⊢ (TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
7	1, 2, 4, 5, 6	prdstopn 22236	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (∏t‘(TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
8		topnfn 16699	. . . . . . . 8 ⊢ TopOpen Fn V
9		dffn2 6516	. . . . . . . . 9 ⊢ ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ↔ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}:2o⟶V)
10	5, 9	sylib 220	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}:2o⟶V)
11		fnfco 6543	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}:2o⟶V) → (TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) Fn 2o)
12	8, 10, 11	sylancr 589	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) Fn 2o)
13		xpsfeq 16836	. . . . . . 7 ⊢ ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) Fn 2o → {⟨∅, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅)⟩, ⟨1o, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o)⟩} = (TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {⟨∅, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅)⟩, ⟨1o, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o)⟩} = (TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
15		0ex 5211	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
16	15	prid1 4698	. . . . . . . . . . 11 ⊢ ∅ ∈ {∅, 1o}
17		df2o3 8117	. . . . . . . . . . 11 ⊢ 2o = {∅, 1o}
18	16, 17	eleqtrri 2912	. . . . . . . . . 10 ⊢ ∅ ∈ 2o
19		fvco2 6758	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ ∅ ∈ 2o) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅) = (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)))
20	5, 18, 19	sylancl 588	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅) = (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)))
21		fvpr0o 16832	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ TopSp → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
22	21	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅) = 𝑅)
23	22	fveq2d 6674	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)) = (TopOpen‘𝑅))
24		xpstopn.j	. . . . . . . . . 10 ⊢ 𝐽 = (TopOpen‘𝑅)
25	23, 24	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘∅)) = 𝐽)
26	20, 25	eqtrd 2856	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅) = 𝐽)
27	26	opeq2d 4810	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ⟨∅, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅)⟩ = ⟨∅, 𝐽⟩)
28		1oex 8110	. . . . . . . . . . . 12 ⊢ 1o ∈ V
29	28	prid2 4699	. . . . . . . . . . 11 ⊢ 1o ∈ {∅, 1o}
30	29, 17	eleqtrri 2912	. . . . . . . . . 10 ⊢ 1o ∈ 2o
31		fvco2 6758	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩} Fn 2o ∧ 1o ∈ 2o) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o) = (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)))
32	5, 30, 31	sylancl 588	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o) = (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)))
33		fvpr1o 16833	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ TopSp → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
34	33	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o) = 𝑆)
35	34	fveq2d 6674	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)) = (TopOpen‘𝑆))
36		xpstopn.k	. . . . . . . . . 10 ⊢ 𝐾 = (TopOpen‘𝑆)
37	35, 36	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘({⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}‘1o)) = 𝐾)
38	32, 37	eqtrd 2856	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o) = 𝐾)
39	38	opeq2d 4810	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ⟨1o, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o)⟩ = ⟨1o, 𝐾⟩)
40	27, 39	preq12d 4677	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {⟨∅, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘∅)⟩, ⟨1o, ((TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})‘1o)⟩} = {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})
41	14, 40	eqtr3d 2858	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) = {⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})
42	41	fveq2d 6674	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (∏t‘(TopOpen ∘ {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}))
43	7, 42	eqtrd 2856	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) = (∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}))
44	43	oveq1d 7171	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) qTop ◡𝐹) = ((∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}) qTop ◡𝐹))
45		xpstps.t	. . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆)
46		xpstopnlem.x	. . . 4 ⊢ 𝑋 = (Base‘𝑅)
47		xpstopnlem.y	. . . 4 ⊢ 𝑌 = (Base‘𝑆)
48		simpl 485	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp)
49		simpr 487	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp)
50		xpstopnlem.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
51		eqid 2821	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
52	45, 46, 47, 48, 49, 50, 51, 1	xpsval 16843	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 = (◡𝐹 “s ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
53	45, 46, 47, 48, 49, 50, 51, 1	xpsrnbas 16844	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ran 𝐹 = (Base‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})))
54	50	xpsff1o2 16842	. . . . 5 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
55		f1ocnv 6627	. . . . 5 ⊢ (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
56	54, 55	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
57		f1ofo 6622	. . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
58	56, 57	syl 17	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
59		ovexd 7191	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}) ∈ V)
60		xpstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑇)
61	52, 53, 58, 59, 6, 60	imastopn 22328	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = ((TopOpen‘((Scalar‘𝑅)Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})) qTop ◡𝐹))
62	46, 24	istps 21542	. . . . 5 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
63	48, 62	sylib 220	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐽 ∈ (TopOn‘𝑋))
64	47, 36	istps 21542	. . . . 5 ⊢ (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
65	49, 64	sylib 220	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐾 ∈ (TopOn‘𝑌))
66	50, 63, 65	xpstopnlem1 22417	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})))
67		hmeocnv 22370	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})) → ◡𝐹 ∈ ((∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})Homeo(𝐽 ×t 𝐾)))
68		hmeoqtop 22383	. . 3 ⊢ (◡𝐹 ∈ ((∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩})Homeo(𝐽 ×t 𝐾)) → (𝐽 ×t 𝐾) = ((∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}) qTop ◡𝐹))
69	66, 67, 68	3syl 18	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (𝐽 ×t 𝐾) = ((∏t‘{⟨∅, 𝐽⟩, ⟨1o, 𝐾⟩}) qTop ◡𝐹))
70	44, 61, 69	3eqtr4d 2866	1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  {cpr 4569  ⟨cop 4573   × cxp 5553  ^◡ccnv 5554  ran crn 5556   ∘ ccom 5559  Oncon0 6191   Fn wfn 6350  ⟶wf 6351  –onto→wfo 6353  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1_oc1o 8095  2_oc2o 8096  Basecbs 16483  Scalarcsca 16568  TopOpenctopn 16695  ∏_tcpt 16712  X_scprds 16719   qTop cqtop 16776   ×_s cxps 16779  TopOnctopon 21518  TopSpctps 21540   ×_t ctx 22168  Homeochmeo 22361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-map 8408  df-ixp 8462  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fi 8875  df-sup 8906  df-inf 8907  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-hom 16589  df-cco 16590  df-rest 16696  df-topn 16697  df-topgen 16717  df-pt 16718  df-prds 16721  df-qtop 16780  df-imas 16781  df-xps 16783  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-cn 21835  df-cnp 21836  df-tx 22170  df-hmeo 22363

This theorem is referenced by:  xpstopn  22420