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Theorem prdstgpd 21838
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 21792 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3587 . . . 4 TopGrp ⊆ Grp
7 fss 6013 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 693 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17446 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 21793 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3587 . . . 4 TopGrp ⊆ TopMnd
12 fss 6013 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 693 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 21837 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2621 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
16 eqid 2621 . . . . . . . 8 (invg𝑌) = (invg𝑌)
1715, 16grpinvf 17387 . . . . . . 7 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
189, 17syl 17 . . . . . 6 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
1918feqmptd 6206 . . . . 5 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
202adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
213adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
228adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
23 simpr 477 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
241, 20, 21, 22, 15, 16, 23prdsinvgd 17447 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
2524mpteq2dva 4704 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
2619, 25eqtrd 2655 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
27 eqid 2621 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
28 eqid 2621 . . . . . . 7 (TopOpen‘𝑌) = (TopOpen‘𝑌)
2928, 15tmdtopon 21795 . . . . . 6 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
3014, 29syl 17 . . . . 5 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
31 topnfn 16007 . . . . . . 7 TopOpen Fn V
32 ffn 6002 . . . . . . . . 9 (𝑅:𝐼⟶TopGrp → 𝑅 Fn 𝐼)
334, 32syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
34 dffn2 6004 . . . . . . . 8 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3533, 34sylib 208 . . . . . . 7 (𝜑𝑅:𝐼⟶V)
36 fnfco 6026 . . . . . . 7 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
3731, 35, 36sylancr 694 . . . . . 6 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
38 fvco3 6232 . . . . . . . . 9 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
394, 38sylan 488 . . . . . . . 8 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
404ffvelrnda 6315 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
41 eqid 2621 . . . . . . . . . 10 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
42 eqid 2621 . . . . . . . . . 10 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
4341, 42tgptopon 21796 . . . . . . . . 9 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
44 topontop 20641 . . . . . . . . 9 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4540, 43, 443syl 18 . . . . . . . 8 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4639, 45eqeltrd 2698 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
4746ralrimiva 2960 . . . . . 6 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
48 ffnfv 6343 . . . . . 6 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
4937, 47, 48sylanbrc 697 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5030adantr 481 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
511, 3, 2, 33, 28prdstopn 21341 . . . . . . . . . . . . 13 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5251adantr 481 . . . . . . . . . . . 12 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5352eqcomd 2627 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
5453, 50eqeltrd 2698 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
55 toponuni 20642 . . . . . . . . . 10 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
56 mpteq1 4697 . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼𝑊)
5949adantr 481 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
60 simpr 477 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦𝐼)
61 eqid 2621 . . . . . . . . . . 11 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
6261, 27ptpjcn 21324 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6358, 59, 60, 62syl3anc 1323 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6457, 63eqeltrd 2698 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6553, 39oveq12d 6622 . . . . . . . 8 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6664, 65eleqtrd 2700 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
67 eqid 2621 . . . . . . . . 9 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
6841, 67tgpinv 21799 . . . . . . . 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 21377 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7139oveq2d 6620 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7270, 71eleqtrrd 2701 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
7327, 30, 2, 49, 72ptcn 21340 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7426, 73eqeltrd 2698 . . 3 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7551oveq2d 6620 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7674, 75eleqtrrd 2701 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7728, 16istgp 21791 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
789, 14, 76, 77syl3anbrc 1244 1 (𝜑𝑌 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555   cuni 4402  cmpt 4673  ccom 5078   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  TopOpenctopn 16003  tcpt 16020  Xscprds 16027  Grpcgrp 17343  invgcminusg 17344  Topctop 20617  TopOnctopon 20618   Cn ccn 20938  TopMndctmd 21784  TopGrpctgp 21785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-sup 8292  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-topgen 16025  df-pt 16026  df-prds 16029  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-minusg 17347  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-cnp 20942  df-tx 21275  df-tmd 21786  df-tgp 21787
This theorem is referenced by: (None)
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