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Theorem prdstgpd 22121
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 (𝜑𝐼𝑊)
prdstgpd.s (𝜑𝑆𝑉)
prdstgpd.r (𝜑𝑅:𝐼⟶TopGrp)
Assertion
Ref Expression
prdstgpd (𝜑𝑌 ∈ TopGrp)

Proof of Theorem prdstgpd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdstgpd.y . . 3 𝑌 = (𝑆Xs𝑅)
2 prdstgpd.i . . 3 (𝜑𝐼𝑊)
3 prdstgpd.s . . 3 (𝜑𝑆𝑉)
4 prdstgpd.r . . . 4 (𝜑𝑅:𝐼⟶TopGrp)
5 tgpgrp 22075 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3740 . . . 4 TopGrp ⊆ Grp
7 fss 6209 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 697 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17718 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 22076 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3740 . . . 4 TopGrp ⊆ TopMnd
12 fss 6209 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 697 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 22120 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2752 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
16 eqid 2752 . . . . . . . 8 (invg𝑌) = (invg𝑌)
1715, 16grpinvf 17659 . . . . . . 7 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
189, 17syl 17 . . . . . 6 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
1918feqmptd 6403 . . . . 5 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
202adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
213adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
228adantr 472 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
23 simpr 479 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
241, 20, 21, 22, 15, 16, 23prdsinvgd 17719 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
2524mpteq2dva 4888 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
2619, 25eqtrd 2786 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
27 eqid 2752 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
28 eqid 2752 . . . . . . 7 (TopOpen‘𝑌) = (TopOpen‘𝑌)
2928, 15tmdtopon 22078 . . . . . 6 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
3014, 29syl 17 . . . . 5 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
31 topnfn 16280 . . . . . . 7 TopOpen Fn V
32 ffn 6198 . . . . . . . . 9 (𝑅:𝐼⟶TopGrp → 𝑅 Fn 𝐼)
334, 32syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
34 dffn2 6200 . . . . . . . 8 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3533, 34sylib 208 . . . . . . 7 (𝜑𝑅:𝐼⟶V)
36 fnfco 6222 . . . . . . 7 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
3731, 35, 36sylancr 698 . . . . . 6 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
38 fvco3 6429 . . . . . . . . 9 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
394, 38sylan 489 . . . . . . . 8 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
404ffvelrnda 6514 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
41 eqid 2752 . . . . . . . . . 10 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
42 eqid 2752 . . . . . . . . . 10 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
4341, 42tgptopon 22079 . . . . . . . . 9 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
44 topontop 20912 . . . . . . . . 9 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4540, 43, 443syl 18 . . . . . . . 8 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4639, 45eqeltrd 2831 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
4746ralrimiva 3096 . . . . . 6 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
48 ffnfv 6543 . . . . . 6 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
4937, 47, 48sylanbrc 701 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5030adantr 472 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
511, 3, 2, 33, 28prdstopn 21625 . . . . . . . . . . . . 13 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5251adantr 472 . . . . . . . . . . . 12 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5352eqcomd 2758 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
5453, 50eqeltrd 2831 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
55 toponuni 20913 . . . . . . . . . 10 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
56 mpteq1 4881 . . . . . . . . . 10 ((Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
5754, 55, 563syl 18 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
582adantr 472 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼𝑊)
5949adantr 472 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
60 simpr 479 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦𝐼)
61 eqid 2752 . . . . . . . . . . 11 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
6261, 27ptpjcn 21608 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6358, 59, 60, 62syl3anc 1473 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6457, 63eqeltrd 2831 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6553, 39oveq12d 6823 . . . . . . . 8 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6664, 65eleqtrd 2833 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
67 eqid 2752 . . . . . . . . 9 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
6841, 67tgpinv 22082 . . . . . . . 8 ((𝑅𝑦) ∈ TopGrp → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
6940, 68syl 17 . . . . . . 7 ((𝜑𝑦𝐼) → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
7050, 66, 69cnmpt11f 21661 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7139oveq2d 6821 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7270, 71eleqtrrd 2834 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
7327, 30, 2, 49, 72ptcn 21624 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7426, 73eqeltrd 2831 . . 3 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7551oveq2d 6821 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7674, 75eleqtrrd 2834 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7728, 16istgp 22074 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
789, 14, 76, 77syl3anbrc 1426 1 (𝜑𝑌 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wcel 2131  wral 3042  Vcvv 3332  wss 3707   cuni 4580  cmpt 4873  ccom 5262   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  Basecbs 16051  TopOpenctopn 16276  tcpt 16293  Xscprds 16300  Grpcgrp 17615  invgcminusg 17616  Topctop 20892  TopOnctopon 20909   Cn ccn 21222  TopMndctmd 22067  TopGrpctgp 22068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-oadd 7725  df-er 7903  df-map 8017  df-ixp 8067  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-sup 8505  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-3 11264  df-4 11265  df-5 11266  df-6 11267  df-7 11268  df-8 11269  df-9 11270  df-n0 11477  df-z 11562  df-dec 11678  df-uz 11872  df-fz 12512  df-struct 16053  df-ndx 16054  df-slot 16055  df-base 16057  df-plusg 16148  df-mulr 16149  df-sca 16151  df-vsca 16152  df-ip 16153  df-tset 16154  df-ple 16155  df-ds 16158  df-hom 16160  df-cco 16161  df-rest 16277  df-topn 16278  df-0g 16296  df-topgen 16298  df-pt 16299  df-prds 16302  df-plusf 17434  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619  df-top 20893  df-topon 20910  df-topsp 20931  df-bases 20944  df-cn 21225  df-cnp 21226  df-tx 21559  df-tmd 22069  df-tgp 22070
This theorem is referenced by: (None)
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