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Theorem prdstgpd 22648
 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 22602 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3975 . . . 4 TopGrp ⊆ Grp
7 fss 6524 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 586 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 18139 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 22603 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3975 . . . 4 TopGrp ⊆ TopMnd
12 fss 6524 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 586 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 22647 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2826 . . . 4 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
16 eqid 2826 . . . . . 6 (TopOpen‘𝑌) = (TopOpen‘𝑌)
17 eqid 2826 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
1816, 17tmdtopon 22605 . . . . 5 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
1914, 18syl 17 . . . 4 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20 topnfn 16689 . . . . . 6 TopOpen Fn V
214ffnd 6512 . . . . . . 7 (𝜑𝑅 Fn 𝐼)
22 dffn2 6513 . . . . . . 7 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
2321, 22sylib 219 . . . . . 6 (𝜑𝑅:𝐼⟶V)
24 fnfco 6540 . . . . . 6 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
2520, 23, 24sylancr 587 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26 fvco3 6757 . . . . . . . 8 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
274, 26sylan 580 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
284ffvelrnda 6847 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
29 eqid 2826 . . . . . . . . 9 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
30 eqid 2826 . . . . . . . . 9 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
3129, 30tgptopon 22606 . . . . . . . 8 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
32 topontop 21437 . . . . . . . 8 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
3328, 31, 323syl 18 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
3427, 33eqeltrd 2918 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
3534ralrimiva 3187 . . . . 5 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36 ffnfv 6878 . . . . 5 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
3725, 35, 36sylanbrc 583 . . . 4 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
3819adantr 481 . . . . . 6 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
391, 3, 2, 21, 16prdstopn 22152 . . . . . . . . . . . 12 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
4039adantr 481 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
4140eqcomd 2832 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
4241, 38eqeltrd 2918 . . . . . . . . 9 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43 toponuni 21438 . . . . . . . . 9 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
44 mpteq1 5151 . . . . . . . . 9 ((Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
4542, 43, 443syl 18 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) = (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)))
462adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝐼𝑊)
4737adantr 481 . . . . . . . . 9 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
48 simpr 485 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦𝐼)
49 eqid 2826 . . . . . . . . . 10 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
5049, 15ptpjcn 22135 . . . . . . . . 9 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5146, 47, 48, 50syl3anc 1365 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5245, 51eqeltrd 2918 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5341, 27oveq12d 7166 . . . . . . 7 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
5452, 53eleqtrd 2920 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
55 eqid 2826 . . . . . . . 8 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
5629, 55tgpinv 22609 . . . . . . 7 ((𝑅𝑦) ∈ TopGrp → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
5728, 56syl 17 . . . . . 6 ((𝜑𝑦𝐼) → (invg‘(𝑅𝑦)) ∈ ((TopOpen‘(𝑅𝑦)) Cn (TopOpen‘(𝑅𝑦))))
5838, 54, 57cnmpt11f 22188 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
5927oveq2d 7164 . . . . 5 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6058, 59eleqtrrd 2921 . . . 4 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6115, 19, 2, 37, 60ptcn 22151 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
62 eqid 2826 . . . . . . 7 (invg𝑌) = (invg𝑌)
6317, 62grpinvf 18080 . . . . . 6 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
649, 63syl 17 . . . . 5 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
6564feqmptd 6730 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
662adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
673adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
688adantr 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
69 simpr 485 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
701, 66, 67, 68, 17, 62, 69prdsinvgd 18140 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
7170mpteq2dva 5158 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
7265, 71eqtrd 2861 . . 3 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
7339oveq2d 7164 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7461, 72, 733eltr4d 2933 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7516, 62istgp 22601 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
769, 14, 74, 75syl3anbrc 1337 1 (𝜑𝑌 ∈ TopGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  ∀wral 3143  Vcvv 3500   ⊆ wss 3940  ∪ cuni 4837   ↦ cmpt 5143   ∘ ccom 5558   Fn wfn 6347  ⟶wf 6348  ‘cfv 6352  (class class class)co 7148  Basecbs 16473  TopOpenctopn 16685  ∏tcpt 16702  Xscprds 16709  Grpcgrp 18033  invgcminusg 18034  Topctop 21417  TopOnctopon 21434   Cn ccn 21748  TopMndctmd 22594  TopGrpctgp 22595 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-ixp 8451  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-fi 8864  df-sup 8895  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-ip 16573  df-tset 16574  df-ple 16575  df-ds 16577  df-hom 16579  df-cco 16580  df-rest 16686  df-topn 16687  df-0g 16705  df-topgen 16707  df-pt 16708  df-prds 16711  df-plusf 17841  df-mgm 17842  df-sgrp 17890  df-mnd 17901  df-grp 18036  df-minusg 18037  df-top 21418  df-topon 21435  df-topsp 21457  df-bases 21470  df-cn 21751  df-cnp 21752  df-tx 22086  df-tmd 22596  df-tgp 22597 This theorem is referenced by: (None)
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