Theorem xpstopnlem2 21614

Description: Lemma for xpstopn 21615. (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 2622	. . . . 5 ⊢ ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})) = ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))
2		fvexd 6203	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (Scalar‘𝑅) ∈ V)
3		2on 7568	. . . . . 6 ⊢ 2𝑜 ∈ On
4	3	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 2𝑜 ∈ On)
5		xpscfn 16219	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜)
6		eqid 2622	. . . . 5 ⊢ (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆})))
7	1, 2, 4, 5, 6	prdstopn 21431	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))))
8		topnfn 16086	. . . . . . . 8 ⊢ TopOpen Fn V
9		dffn2 6047	. . . . . . . . 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 6069	. . . . . . . 8 ⊢ ((TopOpen Fn V ∧ ◡({𝑅} +𝑐 {𝑆}):2𝑜⟶V) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
12	8, 10, 11	sylancr 695	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) Fn 2𝑜)
13		xpsfeq 16224	. . . . . . 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 4790	. . . . . . . . . . . . 13 ⊢ ∅ ∈ V
16	15	prid1 4297	. . . . . . . . . . . 12 ⊢ ∅ ∈ {∅, 1𝑜}
17		df2o3 7573	. . . . . . . . . . . 12 ⊢ 2𝑜 = {∅, 1𝑜}
18	16, 17	eleqtrri 2700	. . . . . . . . . . 11 ⊢ ∅ ∈ 2𝑜
19		fvco2 6273	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ ∅ ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
20	5, 18, 19	sylancl 694	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)))
21		xpsc0 16220	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
22	21	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘∅) = 𝑅)
23	22	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = (TopOpen‘𝑅))
24		xpstopn.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝑅)
25	23, 24	syl6eqr 2674	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘∅)) = 𝐽)
26	20, 25	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅) = 𝐽)
27	26	sneqd 4189	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} = {𝐽})
28		1on 7567	. . . . . . . . . . . . . 14 ⊢ 1𝑜 ∈ On
29	28	elexi 3213	. . . . . . . . . . . . 13 ⊢ 1𝑜 ∈ V
30	29	prid2 4298	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ {∅, 1𝑜}
31	30, 17	eleqtrri 2700	. . . . . . . . . . 11 ⊢ 1𝑜 ∈ 2𝑜
32		fvco2 6273	. . . . . . . . . . 11 ⊢ ((◡({𝑅} +𝑐 {𝑆}) Fn 2𝑜 ∧ 1𝑜 ∈ 2𝑜) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
33	5, 31, 32	sylancl 694	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)))
34		xpsc1 16221	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ TopSp → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
35	34	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (◡({𝑅} +𝑐 {𝑆})‘1𝑜) = 𝑆)
36	35	fveq2d 6195	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = (TopOpen‘𝑆))
37		xpstopn.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘𝑆)
38	36, 37	syl6eqr 2674	. . . . . . . . . 10 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘(◡({𝑅} +𝑐 {𝑆})‘1𝑜)) = 𝐾)
39	33, 38	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜) = 𝐾)
40	39	sneqd 4189	. . . . . . . 8 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)} = {𝐾})
41	27, 40	oveq12d 6668	. . . . . . 7 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ({𝐽} +𝑐 {𝐾}))
42	41	cnveqd 5298	. . . . . 6 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ◡({((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘∅)} +𝑐 {((TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))‘1𝑜)}) = ◡({𝐽} +𝑐 {𝐾}))
43	14, 42	eqtr3d 2658	. . . . 5 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen ∘ ◡({𝑅} +𝑐 {𝑆})) = ◡({𝐽} +𝑐 {𝐾}))
44	43	fveq2d 6195	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (∏t‘(TopOpen ∘ ◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
45	7, 44	eqtrd 2656	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → (TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) = (∏t‘◡({𝐽} +𝑐 {𝐾})))
46	45	oveq1d 6665	. 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 473	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑅 ∈ TopSp)
51		simpr 477	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑆 ∈ TopSp)
52		xpstopnlem.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
53		eqid 2622	. . . 4 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅)
54	47, 48, 49, 50, 51, 52, 53, 1	xpsval 16232	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑇 = (◡𝐹 “s ((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
55	47, 48, 49, 50, 51, 52, 53, 1	xpslem 16233	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → ran 𝐹 = (Base‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))))
56	52	xpsff1o2 16231	. . . . 5 ⊢ 𝐹:(𝑋 × 𝑌)–1-1-onto→ran 𝐹
57		f1ocnv 6149	. . . . 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 6144	. . . 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 6680	. . 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 21523	. 2 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = ((TopOpen‘((Scalar‘𝑅)Xs◡({𝑅} +𝑐 {𝑆}))) qTop ◡𝐹))
64	48, 24	istps 20738	. . . . 5 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
65	50, 64	sylib 208	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐽 ∈ (TopOn‘𝑋))
66	49, 37	istps 20738	. . . . 5 ⊢ (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑌))
67	51, 66	sylib 208	. . . 4 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐾 ∈ (TopOn‘𝑌))
68	52, 65, 67	xpstopnlem1 21612	. . 3 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))))
69		hmeocnv 21565	. . 3 ⊢ (𝐹 ∈ ((𝐽 ×t 𝐾)Homeo(∏t‘◡({𝐽} +𝑐 {𝐾}))) → ◡𝐹 ∈ ((∏t‘◡({𝐽} +𝑐 {𝐾}))Homeo(𝐽 ×t 𝐾)))
70		hmeoqtop 21578	. . 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 2666	1 ⊢ ((𝑅 ∈ TopSp ∧ 𝑆 ∈ TopSp) → 𝑂 = (𝐽 ×t 𝐾))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915  {csn 4177  {cpr 4179   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ∘ ccom 5118  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1_𝑜c1o 7553  2_𝑜c2o 7554   +_𝑐 ccda 8989  Basecbs 15857  Scalarcsca 15944  TopOpenctopn 16082  ∏_tcpt 16099  X_scprds 16106   qTop cqtop 16163   ×_s cxps 16166  TopOnctopon 20715  TopSpctps 20736   ×_t ctx 21363  Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-topgen 16104  df-pt 16105  df-prds 16108  df-qtop 16167  df-imas 16168  df-xps 16170  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558

This theorem is referenced by:  xpstopn  21615