Theorem prdstgpd 24078

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 24031	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3921	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6673	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 587	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 19015	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 24032	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3921	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6673	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
13	4, 11, 12	sylancl 587	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
14	1, 2, 3, 13	prdstmdd 24077	. 2 ⊢ (𝜑 → 𝑌 ∈ TopMnd)
15		eqid 2735	. . . 4 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
16		eqid 2735	. . . . . 6 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
17		eqid 2735	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	16, 17	tmdtopon 24034	. . . . 5 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
19	14, 18	syl 17	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20		topnfn 17377	. . . . . 6 ⊢ TopOpen Fn V
21	4	ffnd 6658	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
22		dffn2 6659	. . . . . . 7 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
23	21, 22	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V)
24		fnfco 6694	. . . . . 6 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
25	20, 23, 24	sylancr 588	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26		fvco3 6928	. . . . . . . 8 ⊢ ((𝑅:𝐼⟶TopGrp ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
27	4, 26	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
28	4	ffvelcdmda 7025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ TopGrp)
29		eqid 2735	. . . . . . . . 9 ⊢ (TopOpen‘(𝑅‘𝑦)) = (TopOpen‘(𝑅‘𝑦))
30		eqid 2735	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
31	29, 30	tgptopon 24035	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
32		topontop 22866	. . . . . . . 8 ⊢ ((TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
33	28, 31, 32	3syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘(𝑅‘𝑦)) ∈ Top)
34	27, 33	eqeltrd 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
35	34	ralrimiva 3127	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36		ffnfv 7060	. . . . 5 ⊢ ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
37	25, 35, 36	sylanbrc 584	. . . 4 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
38	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
39	1, 3, 2, 21, 16	prdstopn 23581	. . . . . . . . . . . 12 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
41	40	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
42	41, 38	eqeltrd 2835	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43		toponuni 22867	. . . . . . . . 9 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
44		mpteq1 5163	. . . . . . . . 9 ⊢ ((Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
45	42, 43, 44	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) = (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)))
46	2	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑊)
47	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
48		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼)
49		eqid 2735	. . . . . . . . . 10 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
50	49, 15	ptpjcn 23564	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
51	46, 47, 48, 50	syl3anc 1374	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
52	45, 51	eqeltrd 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
53	41, 27	oveq12d 7374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
54	52, 53	eleqtrd 2837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
55		eqid 2735	. . . . . . . 8 ⊢ (invg‘(𝑅‘𝑦)) = (invg‘(𝑅‘𝑦))
56	29, 55	tgpinv 24038	. . . . . . 7 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
57	28, 56	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (invg‘(𝑅‘𝑦)) ∈ ((TopOpen‘(𝑅‘𝑦)) Cn (TopOpen‘(𝑅‘𝑦))))
58	38, 54, 57	cnmpt11f 23617	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
59	27	oveq2d 7372	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
60	58, 59	eleqtrrd 2838	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
61	15, 19, 2, 37, 60	ptcn 23580	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
62		eqid 2735	. . . . . . 7 ⊢ (invg‘𝑌) = (invg‘𝑌)
63	17, 62	grpinvf 18951	. . . . . 6 ⊢ (𝑌 ∈ Grp → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
64	9, 63	syl 17	. . . . 5 ⊢ (𝜑 → (invg‘𝑌):(Base‘𝑌)⟶(Base‘𝑌))
65	64	feqmptd 6897	. . . 4 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)))
66	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝐼 ∈ 𝑊)
67	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑆 ∈ 𝑉)
68	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
69		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
70	1, 66, 67, 68, 17, 62, 69	prdsinvgd 19016	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((invg‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
71	70	mpteq2dva 5167	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
72	65, 71	eqtrd 2770	. . 3 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
73	39	oveq2d 7372	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
74	61, 72, 73	3eltr4d 2850	. 2 ⊢ (𝜑 → (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
75	16, 62	istgp 24030	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
76	9, 14, 74, 75	syl3anbrc 1345	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049  Vcvv 3427   ⊆ wss 3885  ∪ cuni 4840   ↦ cmpt 5155   ∘ ccom 5624   Fn wfn 6482  ⟶wf 6483  ‘cfv 6487  (class class class)co 7356  Basecbs 17168  TopOpenctopn 17373  ∏_tcpt 17390  X_scprds 17397  Grpcgrp 18898  inv_gcminusg 18899  Topctop 22846  TopOnctopon 22863   Cn ccn 23177  TopMndctmd 24023  TopGrpctgp 24024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4841  df-int 4880  df-iun 4925  df-iin 4926  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-er 8632  df-map 8764  df-ixp 8835  df-en 8883  df-dom 8884  df-sdom 8885  df-fin 8886  df-fi 9313  df-sup 9344  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12164  df-2 12233  df-3 12234  df-4 12235  df-5 12236  df-6 12237  df-7 12238  df-8 12239  df-9 12240  df-n0 12427  df-z 12514  df-dec 12634  df-uz 12778  df-fz 13451  df-struct 17106  df-slot 17141  df-ndx 17153  df-base 17169  df-plusg 17222  df-mulr 17223  df-sca 17225  df-vsca 17226  df-ip 17227  df-tset 17228  df-ple 17229  df-ds 17231  df-hom 17233  df-cco 17234  df-rest 17374  df-topn 17375  df-0g 17393  df-topgen 17395  df-pt 17396  df-prds 17399  df-plusf 18596  df-mgm 18597  df-sgrp 18676  df-mnd 18692  df-grp 18901  df-minusg 18902  df-top 22847  df-topon 22864  df-topsp 22886  df-bases 22899  df-cn 23180  df-cnp 23181  df-tx 23515  df-tmd 24025  df-tgp 24026

This theorem is referenced by: (None)