Theorem prdstgpd 22253

Description: The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
prdstgpd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdstgpd.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdstgpd.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdstgpd.r	⊢ (𝜑 → 𝑅:𝐼⟶TopGrp)

Assertion

Ref	Expression
prdstgpd	⊢ (𝜑 → 𝑌 ∈ TopGrp)

Proof of Theorem prdstgpd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdstgpd.y	. . 3 ⊢ 𝑌 = (𝑆Xs𝑅)
2		prdstgpd.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		prdstgpd.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
4		prdstgpd.r	. . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶TopGrp)
5		tgpgrp 22207	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3800	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6267	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 581	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 17838	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 22208	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3800	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6267	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
13	4, 11, 12	sylancl 581	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
14	1, 2, 3, 13	prdstmdd 22252	. 2 ⊢ (𝜑 → 𝑌 ∈ TopMnd)
15		eqid 2797	. . . . . . . 8 ⊢ (Base‘𝑌) = (Base‘𝑌)
16		eqid 2797	. . . . . . . 8 ⊢ (invg‘𝑌) = (invg‘𝑌)
17	15, 16	grpinvf 17779	. . . . . . 7 ⊢ (𝑌 ∈ Grp → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
18	9, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
19	18	feqmptd 6472	. . . . 5 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)))
20	2	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
21	3	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
22	8	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
23		simpr 478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
24	1, 20, 21, 22, 15, 16, 23	prdsinvgd 17839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((invg‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
25	24	mpteq2dva 4935	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
26	19, 25	eqtrd 2831	. . . 4 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
27		eqid 2797	. . . . 5 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
28		eqid 2797	. . . . . . 7 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
29	28, 15	tmdtopon 22210	. . . . . 6 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
30	14, 29	syl 17	. . . . 5 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
31		topnfn 16398	. . . . . . 7 ⊢ TopOpen Fn V
32	4	ffnd 6255	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
33		dffn2 6256	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
34	32, 33	sylib 210	. . . . . . 7 ⊢ (𝜑 → 𝑅:𝐼⟶V)
35		fnfco 6282	. . . . . . 7 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
36	31, 34, 35	sylancr 582	. . . . . 6 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
37		fvco3 6498	. . . . . . . . 9 ⊢ ((𝑅:𝐼⟶TopGrp ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
38	4, 37	sylan 576	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
39	4	ffvelrnda 6583	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ TopGrp)
40		eqid 2797	. . . . . . . . . 10 ⊢ (TopOpen‘(𝑅‘𝑦)) = (TopOpen‘(𝑅‘𝑦))
41		eqid 2797	. . . . . . . . . 10 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
42	40, 41	tgptopon 22211	. . . . . . . . 9 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
43		topontop 21043	. . . . . . . . 9 ⊢ ((TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
44	39, 42, 43	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
45	38, 44	eqeltrd 2876	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
46	45	ralrimiva 3145	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
47		ffnfv 6612	. . . . . 6 ⊢ ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
48	36, 46, 47	sylanbrc 579	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
49	30	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
50	1, 3, 2, 32, 28	prdstopn 21757	. . . . . . . . . . . . 13 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
51	50	adantr 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
52	51	eqcomd 2803	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
53	52, 49	eqeltrd 2876	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
54		toponuni 21044	. . . . . . . . . 10 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
55		mpteq1 4928	. . . . . . . . . 10 ⊢ ((Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
56	53, 54, 55	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
57	2	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑊)
58	48	adantr 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
59		simpr 478	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
60		eqid 2797	. . . . . . . . . . 11 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
61	60, 27	ptpjcn 21740	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
62	57, 58, 59, 61	syl3anc 1491	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
63	56, 62	eqeltrd 2876	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
64	52, 38	oveq12d 6894	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
65	63, 64	eleqtrd 2878	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
66		eqid 2797	. . . . . . . . 9 ⊢ (invg‘(𝑅‘𝑦)) = (invg‘(𝑅‘𝑦))
67	40, 66	tgpinv 22214	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
68	39, 67	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
69	49, 65, 68	cnmpt11f 21793	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
70	38	oveq2d 6892	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
71	69, 70	eleqtrrd 2879	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
72	27, 30, 2, 48, 71	ptcn 21756	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
73	26, 72	eqeltrd 2876	. . 3 ⊢ (𝜑 → (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
74	50	oveq2d 6892	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
75	73, 74	eleqtrrd 2879	. 2 ⊢ (𝜑 → (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
76	28, 16	istgp 22206	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
77	9, 14, 75, 76	syl3anbrc 1444	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

Syntax hints:   → wi 4   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3087  Vcvv 3383   ⊆ wss 3767  ∪ cuni 4626   ↦ cmpt 4920   ∘ ccom 5314   Fn wfn 6094  ⟶wf 6095  ‘cfv 6099  (class class class)co 6876  Basecbs 16181  TopOpenctopn 16394  ∏_tcpt 16411  X_scprds 16418  Grpcgrp 17735  inv_gcminusg 17736  Topctop 21023  TopOnctopon 21040   Cn ccn 21354  TopMndctmd 22199  TopGrpctgp 22200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-er 7980  df-map 8095  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-fi 8557  df-sup 8588  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-3 11373  df-4 11374  df-5 11375  df-6 11376  df-7 11377  df-8 11378  df-9 11379  df-n0 11577  df-z 11663  df-dec 11780  df-uz 11927  df-fz 12577  df-struct 16183  df-ndx 16184  df-slot 16185  df-base 16187  df-plusg 16277  df-mulr 16278  df-sca 16280  df-vsca 16281  df-ip 16282  df-tset 16283  df-ple 16284  df-ds 16286  df-hom 16288  df-cco 16289  df-rest 16395  df-topn 16396  df-0g 16414  df-topgen 16416  df-pt 16417  df-prds 16420  df-plusf 17553  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-top 21024  df-topon 21041  df-topsp 21063  df-bases 21076  df-cn 21357  df-cnp 21358  df-tx 21691  df-tmd 22201  df-tgp 22202

This theorem is referenced by: (None)