Theorem xpstopnlem2 21424

Description: Lemma for xpstopn 21425. (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} +𝑐 {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 2610	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
2		fvex 6113	. . . . . 6 ⊢ (Scalar‘𝑅) ∈ V
3	2	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V)
4		2on 7455	. . . . . 6 ⊢ 2𝑜 ∈ On
5	4	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2𝑜 ∈ On)
6		xpscfn 16042	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
7		eqid 2610	. . . . 5 ⊢ (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
8	1, 3, 5, 6, 7	prdstopn 21241	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))))
9		topnfn 15909	. . . . . . . 8 ⊢ TopOpen Fn V
10		dffn2 5960	. . . . . . . . 9 ⊢ (◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ↔ ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V)
11	6, 10	sylib 207	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V)
12		fnfco 5982	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
13	9, 11, 12	sylancr 694	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
14		xpsfeq 16047	. . . . . . 7 ⊢ ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜 → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})))
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})))
16		0ex 4718	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
17	16	prid1 4241	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1𝑜}
18		df2o3 7460	. . . . . . . . . . . 12 ⊢ 2𝑜 = {∅, 1𝑜}
19	17, 18	eleqtrri 2687	. . . . . . . . . . 11 ⊢ ∅ ∈ 2𝑜
20		fvco2 6183	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
21	6, 19, 20	sylancl 693	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
22		xpsc0 16043	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
23	22	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
24	23	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (TopOpen‘𝑅))
25		xpstopn.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝑅)
26	24, 25	syl6eqr 2662	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = 𝐽)
27	21, 26	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = 𝐽)
28	27	sneqd 4137	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} = {𝐽})
29		1on 7454	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
30	29	elexi 3186	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ V
31	30	prid2 4242	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ {∅, 1𝑜}
32	31, 18	eleqtrri 2687	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ 2𝑜
33		fvco2 6183	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
34	6, 32, 33	sylancl 693	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
35		xpsc1 16044	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
36	35	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
37	36	fveq2d 6107	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (TopOpen‘𝑆))
38		xpstopn.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘𝑆)
39	37, 38	syl6eqr 2662	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = 𝐾)
40	34, 39	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = 𝐾)
41	40	sneqd 4137	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)} = {𝐾})
42	28, 41	oveq12d 6567	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ({𝐽} +𝑐 {𝐾}))
43	42	cnveqd 5220	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ◡({𝐽} +𝑐 {𝐾}))
44	15, 43	eqtr3d 2646	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) = ◡({𝐽} +𝑐 {𝐾}))
45	44	fveq2d 6107	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
46	8, 45	eqtrd 2644	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
47	46	oveq1d 6564	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) qTop ◡𝐹) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
48		xpstps.t	. . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆)
49		xpstopnlem.x	. . . 4 ⊢ 𝑋 = (Base‘𝑅)
50		xpstopnlem.y	. . . 4 ⊢ 𝑌 = (Base‘𝑆)
51		simpl 472	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp)
52		simpr 476	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp)
53		xpstopnlem.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
54		eqid 2610	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
55	48, 49, 50, 51, 52, 53, 54, 1	xpsval 16055	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 = (◡𝐹 “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
56	48, 49, 50, 51, 52, 53, 54, 1	xpslem 16056	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ran 𝐹 = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
57	53	xpsff1o2 16054	. . . . 5 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
58		f1ocnv 6062	. . . . 5 ⊢ (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
59	57, 58	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
60		f1ofo 6057	. . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
61	59, 60	syl 17	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
62		ovex 6577	. . . 4 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V
63	62	a1i 11	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
64		xpstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑇)
65	55, 56, 61, 63, 7, 64	imastopn 21333	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = ((TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) qTop ◡𝐹))
66	49, 25	istps 20551	. . . . 5 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
67	51, 66	sylib 207	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐽 ∈ (TopOn‘𝑋))
68	50, 38	istps 20551	. . . . 5 ⊢ (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
69	52, 68	sylib 207	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐾 ∈ (TopOn‘𝑌))
70	53, 67, 69	xpstopnlem1 21422	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
71		hmeocnv 21375	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))) → ◡𝐹 ∈ ((∏t‘◡({𝐽} +𝑐 {𝐾}))Homeo(𝐽 ×t 𝐾)))
72		hmeoqtop 21388	. . 3 ⊢ (◡𝐹 ∈ ((∏t‘◡({𝐽} +𝑐 {𝐾}))Homeo(𝐽 ×t 𝐾)) → (𝐽 ×t 𝐾) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
73	70, 71, 72	3syl 18	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (𝐽 ×t 𝐾) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
74	47, 65, 73	3eqtr4d 2654	1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127   × cxp 5036  ^◡ccnv 5037  ran crn 5039   ∘ ccom 5042  Oncon0 5640   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1_𝑜c1o 7440  2_𝑜c2o 7441   +_𝑐 ccda 8872  Basecbs 15695  Scalarcsca 15771  TopOpenctopn 15905  ∏_tcpt 15922  X_scprds 15929   qTop cqtop 15986   ×_s cxps 15989  TopOnctopon 20518  TopSpctps 20519   ×_t ctx 21173  Homeochmeo 21366

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  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-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  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-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-rest 15906  df-topn 15907  df-topgen 15927  df-pt 15928  df-prds 15931  df-qtop 15990  df-imas 15991  df-xps 15993  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cn 20841  df-cnp 20842  df-tx 21175  df-hmeo 21368

This theorem is referenced by:  xpstopn  21425