Theorem prdstgpd 24149

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 24102	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
6	5	ssriv 3999	. . . 4 ⊢ TopGrp ⊆ Grp
7		fss 6753	. . . 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 19081	. 2 ⊢ (𝜑 → 𝑌 ∈ Grp)
10		tgptmd 24103	. . . . 5 ⊢ (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
11	10	ssriv 3999	. . . 4 ⊢ TopGrp ⊆ TopMnd
12		fss 6753	. . . 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 24148	. 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 24105	. . . . 5 ⊢ (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
19	14, 18	syl 17	. . . 4 ⊢ (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20		topnfn 17472	. . . . . 6 ⊢ TopOpen Fn V
21	4	ffnd 6738	. . . . . . 7 ⊢ (𝜑 → 𝑅 Fn 𝐼)
22		dffn2 6739	. . . . . . 7 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
23	21, 22	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝑅:𝐼⟶V)
24		fnfco 6774	. . . . . 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 7008	. . . . . . . 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 7104	. . . . . . . 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 24106	. . . . . . . 8 ⊢ ((𝑅‘𝑦) ∈ TopGrp → (TopOpen‘(𝑅‘𝑦)) ∈ (TopOn‘(Base‘(𝑅‘𝑦))))
32		topontop 22935	. . . . . . . 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 2839	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
35	34	ralrimiva 3144	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36		ffnfv 7139	. . . . 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 23652	. . . . . . . . . . . 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 2839	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43		toponuni 22936	. . . . . . . . 9 ⊢ ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = ∪ (∏t‘(TopOpen ∘ 𝑅)))
44		mpteq1 5241	. . . . . . . . 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 23635	. . . . . . . . 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 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ ∪ (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
52	45, 51	eqeltrd 2839	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥‘𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
53	41, 27	oveq12d 7449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
54	52, 53	eleqtrd 2841	. . . . . 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 24109	. . . . . . 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 23688	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
59	27	oveq2d 7447	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅‘𝑦))))
60	58, 59	eleqtrrd 2842	. . . 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 23651	. . 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 19017	. . . . . 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 6977	. . . 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 19082	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑌)) → ((invg‘𝑌)‘𝑥) = (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦))))
71	70	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
72	65, 71	eqtrd 2775	. . 3 ⊢ (𝜑 → (invg‘𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦 ∈ 𝐼 ↦ ((invg‘(𝑅‘𝑦))‘(𝑥‘𝑦)))))
73	39	oveq2d 7447	. . 3 ⊢ (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
74	61, 72, 73	3eltr4d 2854	. 2 ⊢ (𝜑 → (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
75	16, 62	istgp 24101	. 2 ⊢ (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg‘𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
76	9, 14, 74, 75	syl3anbrc 1342	1 ⊢ (𝜑 → 𝑌 ∈ TopGrp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478   ⊆ wss 3963  ∪ cuni 4912   ↦ cmpt 5231   ∘ ccom 5693   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  TopOpenctopn 17468  ∏_tcpt 17485  X_scprds 17492  Grpcgrp 18964  inv_gcminusg 18965  Topctop 22915  TopOnctopon 22932   Cn ccn 23248  TopMndctmd 24094  TopGrpctgp 24095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fi 9449  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-slot 17216  df-ndx 17228  df-base 17246  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-topgen 17490  df-pt 17491  df-prds 17494  df-plusf 18665  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cn 23251  df-cnp 23252  df-tx 23586  df-tmd 24096  df-tgp 24097

This theorem is referenced by: (None)