Theorem xpstopnlem2 21662

Description: Lemma for xpstopn 21663. (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 2651	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
2		fvexd 6241	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V)
3		2on 7613	. . . . . 6 ⊢ 2𝑜 ∈ On
4	3	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2𝑜 ∈ On)
5		xpscfn 16266	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
6		eqid 2651	. . . . 5 ⊢ (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
7	1, 2, 4, 5, 6	prdstopn 21479	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))))
8		topnfn 16133	. . . . . . . 8 ⊢ TopOpen Fn V
9		dffn2 6085	. . . . . . . . 9 ⊢ (◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ↔ ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V)
10	5, 9	sylib 208	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V)
11		fnfco 6107	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
12	8, 10, 11	sylancr 696	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
13		xpsfeq 16271	. . . . . . 7 ⊢ ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜 → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})))
14	12, 13	syl 17	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})))
15		0ex 4823	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
16	15	prid1 4329	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1𝑜}
17		df2o3 7618	. . . . . . . . . . . 12 ⊢ 2𝑜 = {∅, 1𝑜}
18	16, 17	eleqtrri 2729	. . . . . . . . . . 11 ⊢ ∅ ∈ 2𝑜
19		fvco2 6312	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
20	5, 18, 19	sylancl 695	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
21		xpsc0 16267	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
22	21	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
23	22	fveq2d 6233	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (TopOpen‘𝑅))
24		xpstopn.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝑅)
25	23, 24	syl6eqr 2703	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = 𝐽)
26	20, 25	eqtrd 2685	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = 𝐽)
27	26	sneqd 4222	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} = {𝐽})
28		1on 7612	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
29	28	elexi 3244	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ V
30	29	prid2 4330	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ {∅, 1𝑜}
31	30, 17	eleqtrri 2729	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ 2𝑜
32		fvco2 6312	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
33	5, 31, 32	sylancl 695	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
34		xpsc1 16268	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
35	34	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
36	35	fveq2d 6233	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (TopOpen‘𝑆))
37		xpstopn.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘𝑆)
38	36, 37	syl6eqr 2703	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = 𝐾)
39	33, 38	eqtrd 2685	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = 𝐾)
40	39	sneqd 4222	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)} = {𝐾})
41	27, 40	oveq12d 6708	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ({𝐽} +𝑐 {𝐾}))
42	41	cnveqd 5330	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ◡({𝐽} +𝑐 {𝐾}))
43	14, 42	eqtr3d 2687	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) = ◡({𝐽} +𝑐 {𝐾}))
44	43	fveq2d 6233	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
45	7, 44	eqtrd 2685	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
46	45	oveq1d 6705	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) qTop ◡𝐹) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
47		xpstps.t	. . . 4 ⊢ 𝑇 = (𝑅 ×s 𝑆)
48		xpstopnlem.x	. . . 4 ⊢ 𝑋 = (Base‘𝑅)
49		xpstopnlem.y	. . . 4 ⊢ 𝑌 = (Base‘𝑆)
50		simpl 472	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp)
51		simpr 476	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp)
52		xpstopnlem.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
53		eqid 2651	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
54	47, 48, 49, 50, 51, 52, 53, 1	xpsval 16279	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 = (◡𝐹 “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
55	47, 48, 49, 50, 51, 52, 53, 1	xpslem 16280	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ran 𝐹 = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
56	52	xpsff1o2 16278	. . . . 5 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
57		f1ocnv 6187	. . . . 5 ⊢ (𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹 → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
58	56, 57	mp1i 13	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌))
59		f1ofo 6182	. . . 4 ⊢ (◡𝐹:ran 𝐹–1-1-onto→(𝑋 × 𝑌) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
60	58, 59	syl 17	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡𝐹:ran 𝐹–onto→(𝑋 × 𝑌))
61		ovexd 6720	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) ∈ V)
62		xpstopn.o	. . 3 ⊢ 𝑂 = (TopOpen‘𝑇)
63	54, 55, 60, 61, 6, 62	imastopn 21571	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = ((TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) qTop ◡𝐹))
64	48, 24	istps 20786	. . . . 5 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
65	50, 64	sylib 208	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐽 ∈ (TopOn‘𝑋))
66	49, 37	istps 20786	. . . . 5 ⊢ (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
67	51, 66	sylib 208	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐾 ∈ (TopOn‘𝑌))
68	52, 65, 67	xpstopnlem1 21660	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
69		hmeocnv 21613	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))) → ◡𝐹 ∈ ((∏t‘◡({𝐽} +𝑐 {𝐾}))Homeo(𝐽 ×t 𝐾)))
70		hmeoqtop 21626	. . 3 ⊢ (◡𝐹 ∈ ((∏t‘◡({𝐽} +𝑐 {𝐾}))Homeo(𝐽 ×t 𝐾)) → (𝐽 ×t 𝐾) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
71	68, 69, 70	3syl 18	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (𝐽 ×t 𝐾) = ((∏t‘◡({𝐽} +𝑐 {𝐾})) qTop ◡𝐹))
72	46, 63, 71	3eqtr4d 2695	1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  {csn 4210  {cpr 4212   × cxp 5141  ^◡ccnv 5142  ran crn 5144   ∘ ccom 5147  Oncon0 5761   Fn wfn 5921  ⟶wf 5922  –onto→wfo 5924  –1-1-onto→wf1o 5925  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  1_𝑜c1o 7598  2_𝑜c2o 7599   +_𝑐 ccda 9027  Basecbs 15904  Scalarcsca 15991  TopOpenctopn 16129  ∏_tcpt 16146  X_scprds 16153   qTop cqtop 16210   ×_s cxps 16213  TopOnctopon 20763  TopSpctps 20784   ×_t ctx 21411  Homeochmeo 21604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-topgen 16151  df-pt 16152  df-prds 16155  df-qtop 16214  df-imas 16215  df-xps 16217  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-cnp 21080  df-tx 21413  df-hmeo 21606

This theorem is referenced by:  xpstopn  21663