MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prdstgpd Structured version   Visualization version   GIF version

Theorem prdstgpd 21975
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 21929 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3640 . . . 4 TopGrp ⊆ Grp
7 fss 6094 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 695 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17572 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 21930 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3640 . . . 4 TopGrp ⊆ TopMnd
12 fss 6094 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 695 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 21974 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2651 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
16 eqid 2651 . . . . . . . 8 (invg𝑌) = (invg𝑌)
1715, 16grpinvf 17513 . . . . . . 7 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
189, 17syl 17 . . . . . 6 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
1918feqmptd 6288 . . . . 5 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
202adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
213adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
228adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
23 simpr 476 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
241, 20, 21, 22, 15, 16, 23prdsinvgd 17573 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
2524mpteq2dva 4777 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
2619, 25eqtrd 2685 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
27 eqid 2651 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
28 eqid 2651 . . . . . . 7 (TopOpen‘𝑌) = (TopOpen‘𝑌)
2928, 15tmdtopon 21932 . . . . . 6 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
3014, 29syl 17 . . . . 5 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
31 topnfn 16133 . . . . . . 7 TopOpen Fn V
32 ffn 6083 . . . . . . . . 9 (𝑅:𝐼⟶TopGrp → 𝑅 Fn 𝐼)
334, 32syl 17 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
34 dffn2 6085 . . . . . . . 8 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3533, 34sylib 208 . . . . . . 7 (𝜑𝑅:𝐼⟶V)
36 fnfco 6107 . . . . . . 7 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
3731, 35, 36sylancr 696 . . . . . 6 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
38 fvco3 6314 . . . . . . . . 9 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
394, 38sylan 487 . . . . . . . 8 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
404ffvelrnda 6399 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
41 eqid 2651 . . . . . . . . . 10 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
42 eqid 2651 . . . . . . . . . 10 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
4341, 42tgptopon 21933 . . . . . . . . 9 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
44 topontop 20766 . . . . . . . . 9 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4540, 43, 443syl 18 . . . . . . . 8 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4639, 45eqeltrd 2730 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
4746ralrimiva 2995 . . . . . 6 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
48 ffnfv 6428 . . . . . 6 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
4937, 47, 48sylanbrc 699 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
5030adantr 480 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
511, 3, 2, 33, 28prdstopn 21479 . . . . . . . . . . . . 13 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5251adantr 480 . . . . . . . . . . . 12 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5352eqcomd 2657 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
5453, 50eqeltrd 2730 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
55 toponuni 20767 . . . . . . . . . 10 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
56 mpteq1 4770 . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼𝑊)
5949adantr 480 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
60 simpr 476 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦𝐼)
61 eqid 2651 . . . . . . . . . . 11 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
6261, 27ptpjcn 21462 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6358, 59, 60, 62syl3anc 1366 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6457, 63eqeltrd 2730 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6553, 39oveq12d 6708 . . . . . . . 8 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6664, 65eleqtrd 2732 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
67 eqid 2651 . . . . . . . . 9 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
6841, 67tgpinv 21936 . . . . . . . 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 21515 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7139oveq2d 6706 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7270, 71eleqtrrd 2733 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
7327, 30, 2, 49, 72ptcn 21478 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7426, 73eqeltrd 2730 . . 3 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7551oveq2d 6706 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7674, 75eleqtrrd 2733 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7728, 16istgp 21928 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
789, 14, 76, 77syl3anbrc 1265 1 (𝜑𝑌 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607   cuni 4468  cmpt 4762  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  TopOpenctopn 16129  tcpt 16146  Xscprds 16153  Grpcgrp 17469  invgcminusg 17470  Topctop 20746  TopOnctopon 20763   Cn ccn 21076  TopMndctmd 21921  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-topgen 16151  df-pt 16152  df-prds 16155  df-plusf 17288  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cn 21079  df-cnp 21080  df-tx 21413  df-tmd 21923  df-tgp 21924
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator