Theorem prdstgpd 24068

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 24021	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3967	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6727	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
8	4, 6, 7	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶Grp)
9	1, 2, 3, 8	prdsgrpd 19038	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 24022	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3967	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6727	. . . 4 ⊢ ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
13	4, 11, 12	sylancl 586	. . 3 ⊢ (𝜑 → 𝑅:𝐼⟶TopMnd)
14	1, 2, 3, 13	prdstmdd 24067	. 2 ⊢ (𝜑 → 𝑌 ∈ TopMnd)
15		eqid 2736	. . . 4 ⊢ (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
16		eqid 2736	. . . . . 6 ⊢ (TopOpen‘𝑌) = (TopOpen‘𝑌)
17		eqid 2736	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
18	16, 17	tmdtopon 24024	. . . . 5 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
19	14, 18	syl 17	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20		topnfn 17444	. . . . . 6 ⊢ TopOpen Fn V
21	4	ffnd 6712	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
22		dffn2 6713	. . . . . . 7 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
23	21, 22	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V)
24		fnfco 6748	. . . . . 6 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
25	20, 23, 24	sylancr 587	. . . . 5 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26		fvco3 6983	. . . . . . . 8 ⊢ ((𝑅:𝐼⟶TopGrp ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
27	4, 26	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅‘𝑦)))
28	4	ffvelcdmda 7079	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑅‘𝑦) ∈ TopGrp)
29		eqid 2736	. . . . . . . . 9 ⊢ (TopOpen‘(𝑅‘𝑦)) = (TopOpen‘(𝑅‘𝑦))
30		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(𝑅‘𝑦)) = (Base‘(𝑅‘𝑦))
31	29, 30	tgptopon 24025	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
32		topontop 22856	. . . . . . . 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 3133	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36		ffnfv 7114	. . . . 5 ⊢ ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
37	25, 35, 36	sylanbrc 583	. . . 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 23571	. . . . . . . . . . . 12 ⊢ (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
40	39	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
41	40	eqcomd 2742	. . . . . . . . . 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 22857	. . . . . . . . 9 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
44		mpteq1 5214	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ ∪ (∏t‘(TopOpen ∘ 𝑅)) = ∪ (∏t‘(TopOpen ∘ 𝑅))
50	49, 15	ptpjcn 23554	. . . . . . . . 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 1373	. . . . . . . 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 7428	. . . . . . 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 2736	. . . . . . . 8 ⊢ (invg‘(𝑅‘𝑦)) = (invg‘(𝑅‘𝑦))
56	29, 55	tgpinv 24028	. . . . . . 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 23607	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
59	27	oveq2d 7426	. . . . 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 23570	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
62		eqid 2736	. . . . . . 7 ⊢ (invg‘𝑌) = (invg‘𝑌)
63	17, 62	grpinvf 18974	. . . . . 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 6952	. . . 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 19039	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((invg‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
71	70	mpteq2dva 5219	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
72	65, 71	eqtrd 2771	. . 3 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
73	39	oveq2d 7426	. . 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 24020	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
76	9, 14, 74, 75	syl3anbrc 1344	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  ∪ cuni 4888   ↦ cmpt 5206   ∘ ccom 5663   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  TopOpenctopn 17440  ∏_tcpt 17457  X_scprds 17464  Grpcgrp 18921  inv_gcminusg 18922  Topctop 22836  TopOnctopon 22853   Cn ccn 23167  TopMndctmd 24013  TopGrpctgp 24014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-iin 4975  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8724  df-map 8847  df-ixp 8917  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9428  df-sup 9459  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-plusg 17289  df-mulr 17290  df-sca 17292  df-vsca 17293  df-ip 17294  df-tset 17295  df-ple 17296  df-ds 17298  df-hom 17300  df-cco 17301  df-rest 17441  df-topn 17442  df-0g 17460  df-topgen 17462  df-pt 17463  df-prds 17466  df-plusf 18622  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925  df-top 22837  df-topon 22854  df-topsp 22876  df-bases 22889  df-cn 23170  df-cnp 23171  df-tx 23505  df-tmd 24015  df-tgp 24016

This theorem is referenced by: (None)