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

Theorem prdstgpd 22253
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 22207 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3800 . . . 4 TopGrp ⊆ Grp
7 fss 6267 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 581 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 17838 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 22208 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3800 . . . 4 TopGrp ⊆ TopMnd
12 fss 6267 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 581 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 22252 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2797 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
16 eqid 2797 . . . . . . . 8 (invg𝑌) = (invg𝑌)
1715, 16grpinvf 17779 . . . . . . 7 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
189, 17syl 17 . . . . . 6 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
1918feqmptd 6472 . . . . 5 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
202adantr 473 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
213adantr 473 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
228adantr 473 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
23 simpr 478 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
241, 20, 21, 22, 15, 16, 23prdsinvgd 17839 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
2524mpteq2dva 4935 . . . . 5 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
2619, 25eqtrd 2831 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
27 eqid 2797 . . . . 5 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
28 eqid 2797 . . . . . . 7 (TopOpen‘𝑌) = (TopOpen‘𝑌)
2928, 15tmdtopon 22210 . . . . . 6 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
3014, 29syl 17 . . . . 5 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
31 topnfn 16398 . . . . . . 7 TopOpen Fn V
324ffnd 6255 . . . . . . . 8 (𝜑𝑅 Fn 𝐼)
33 dffn2 6256 . . . . . . . 8 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
3432, 33sylib 210 . . . . . . 7 (𝜑𝑅:𝐼⟶V)
35 fnfco 6282 . . . . . . 7 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
3631, 34, 35sylancr 582 . . . . . 6 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
37 fvco3 6498 . . . . . . . . 9 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
384, 37sylan 576 . . . . . . . 8 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
394ffvelrnda 6583 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
40 eqid 2797 . . . . . . . . . 10 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
41 eqid 2797 . . . . . . . . . 10 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
4240, 41tgptopon 22211 . . . . . . . . 9 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
43 topontop 21043 . . . . . . . . 9 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4439, 42, 433syl 18 . . . . . . . 8 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
4538, 44eqeltrd 2876 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
4645ralrimiva 3145 . . . . . 6 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
47 ffnfv 6612 . . . . . 6 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
4836, 46, 47sylanbrc 579 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
4930adantr 473 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
501, 3, 2, 32, 28prdstopn 21757 . . . . . . . . . . . . 13 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5150adantr 473 . . . . . . . . . . . 12 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
5251eqcomd 2803 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
5352, 49eqeltrd 2876 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
54 toponuni 21044 . . . . . . . . . 10 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
55 mpteq1 4928 . . . . . . . . . 10 ((Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
5653, 54, 553syl 18 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
572adantr 473 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝐼𝑊)
5848adantr 473 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
59 simpr 478 . . . . . . . . . 10 ((𝜑𝑦𝐼) → 𝑦𝐼)
60 eqid 2797 . . . . . . . . . . 11 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
6160, 27ptpjcn 21740 . . . . . . . . . 10 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6257, 58, 59, 61syl3anc 1491 . . . . . . . . 9 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6356, 62eqeltrd 2876 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6452, 38oveq12d 6894 . . . . . . . 8 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6563, 64eleqtrd 2878 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
66 eqid 2797 . . . . . . . . 9 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
6740, 66tgpinv 22214 . . . . . . . 8 ((𝑅𝑦) ∈ TopGrp → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
6839, 67syl 17 . . . . . . 7 ((𝜑𝑦𝐼) → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
6949, 65, 68cnmpt11f 21793 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7038oveq2d 6892 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
7169, 70eleqtrrd 2879 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
7227, 30, 2, 48, 71ptcn 21756 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7326, 72eqeltrd 2876 . . 3 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7450oveq2d 6892 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7573, 74eleqtrrd 2879 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7628, 16istgp 22206 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
779, 14, 75, 76syl3anbrc 1444 1 (𝜑𝑌 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  Vcvv 3383  wss 3767   cuni 4626  cmpt 4920  ccom 5314   Fn wfn 6094  wf 6095  cfv 6099  (class class class)co 6876  Basecbs 16181  TopOpenctopn 16394  tcpt 16411  Xscprds 16418  Grpcgrp 17735  invgcminusg 17736  Topctop 21023  TopOnctopon 21040   Cn ccn 21354  TopMndctmd 22199  TopGrpctgp 22200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-1cn 10280  ax-icn 10281  ax-addcl 10282  ax-addrcl 10283  ax-mulcl 10284  ax-mulrcl 10285  ax-mulcom 10286  ax-addass 10287  ax-mulass 10288  ax-distr 10289  ax-i2m1 10290  ax-1ne0 10291  ax-1rid 10292  ax-rnegex 10293  ax-rrecex 10294  ax-cnre 10295  ax-pre-lttri 10296  ax-pre-lttrn 10297  ax-pre-ltadd 10298  ax-pre-mulgt0 10299
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-pred 5896  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-om 7298  df-1st 7399  df-2nd 7400  df-wrecs 7643  df-recs 7705  df-rdg 7743  df-1o 7797  df-oadd 7801  df-er 7980  df-map 8095  df-ixp 8147  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-fi 8557  df-sup 8588  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-sub 10556  df-neg 10557  df-nn 11311  df-2 11372  df-3 11373  df-4 11374  df-5 11375  df-6 11376  df-7 11377  df-8 11378  df-9 11379  df-n0 11577  df-z 11663  df-dec 11780  df-uz 11927  df-fz 12577  df-struct 16183  df-ndx 16184  df-slot 16185  df-base 16187  df-plusg 16277  df-mulr 16278  df-sca 16280  df-vsca 16281  df-ip 16282  df-tset 16283  df-ple 16284  df-ds 16286  df-hom 16288  df-cco 16289  df-rest 16395  df-topn 16396  df-0g 16414  df-topgen 16416  df-pt 16417  df-prds 16420  df-plusf 17553  df-mgm 17554  df-sgrp 17596  df-mnd 17607  df-grp 17738  df-minusg 17739  df-top 21024  df-topon 21041  df-topsp 21063  df-bases 21076  df-cn 21357  df-cnp 21358  df-tx 21691  df-tmd 22201  df-tgp 22202
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator