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Theorem prdstgpd 22876
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 22829 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ Grp)
65ssriv 3881 . . . 4 TopGrp ⊆ Grp
7 fss 6521 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ Grp) → 𝑅:𝐼⟶Grp)
84, 6, 7sylancl 589 . . 3 (𝜑𝑅:𝐼⟶Grp)
91, 2, 3, 8prdsgrpd 18327 . 2 (𝜑𝑌 ∈ Grp)
10 tgptmd 22830 . . . . 5 (𝑥 ∈ TopGrp → 𝑥 ∈ TopMnd)
1110ssriv 3881 . . . 4 TopGrp ⊆ TopMnd
12 fss 6521 . . . 4 ((𝑅:𝐼⟶TopGrp ∧ TopGrp ⊆ TopMnd) → 𝑅:𝐼⟶TopMnd)
134, 11, 12sylancl 589 . . 3 (𝜑𝑅:𝐼⟶TopMnd)
141, 2, 3, 13prdstmdd 22875 . 2 (𝜑𝑌 ∈ TopMnd)
15 eqid 2738 . . . 4 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
16 eqid 2738 . . . . . 6 (TopOpen‘𝑌) = (TopOpen‘𝑌)
17 eqid 2738 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
1816, 17tmdtopon 22832 . . . . 5 (𝑌 ∈ TopMnd → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
1914, 18syl 17 . . . 4 (𝜑 → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
20 topnfn 16802 . . . . . 6 TopOpen Fn V
214ffnd 6505 . . . . . . 7 (𝜑𝑅 Fn 𝐼)
22 dffn2 6506 . . . . . . 7 (𝑅 Fn 𝐼𝑅:𝐼⟶V)
2321, 22sylib 221 . . . . . 6 (𝜑𝑅:𝐼⟶V)
24 fnfco 6543 . . . . . 6 ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
2520, 23, 24sylancr 590 . . . . 5 (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
26 fvco3 6767 . . . . . . . 8 ((𝑅:𝐼⟶TopGrp ∧ 𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
274, 26sylan 583 . . . . . . 7 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) = (TopOpen‘(𝑅𝑦)))
284ffvelrnda 6861 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑅𝑦) ∈ TopGrp)
29 eqid 2738 . . . . . . . . 9 (TopOpen‘(𝑅𝑦)) = (TopOpen‘(𝑅𝑦))
30 eqid 2738 . . . . . . . . 9 (Base‘(𝑅𝑦)) = (Base‘(𝑅𝑦))
3129, 30tgptopon 22833 . . . . . . . 8 ((𝑅𝑦) ∈ TopGrp → (TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))))
32 topontop 21664 . . . . . . . 8 ((TopOpen‘(𝑅𝑦)) ∈ (TopOn‘(Base‘(𝑅𝑦))) → (TopOpen‘(𝑅𝑦)) ∈ Top)
3328, 31, 323syl 18 . . . . . . 7 ((𝜑𝑦𝐼) → (TopOpen‘(𝑅𝑦)) ∈ Top)
3427, 33eqeltrd 2833 . . . . . 6 ((𝜑𝑦𝐼) → ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
3534ralrimiva 3096 . . . . 5 (𝜑 → ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top)
36 ffnfv 6892 . . . . 5 ((TopOpen ∘ 𝑅):𝐼⟶Top ↔ ((TopOpen ∘ 𝑅) Fn 𝐼 ∧ ∀𝑦𝐼 ((TopOpen ∘ 𝑅)‘𝑦) ∈ Top))
3725, 35, 36sylanbrc 586 . . . 4 (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
3819adantr 484 . . . . . 6 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) ∈ (TopOn‘(Base‘𝑌)))
391, 3, 2, 21, 16prdstopn 22379 . . . . . . . . . . . 12 (𝜑 → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
4039adantr 484 . . . . . . . . . . 11 ((𝜑𝑦𝐼) → (TopOpen‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
4140eqcomd 2744 . . . . . . . . . 10 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (TopOpen‘𝑌))
4241, 38eqeltrd 2833 . . . . . . . . 9 ((𝜑𝑦𝐼) → (∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)))
43 toponuni 21665 . . . . . . . . 9 ((∏t‘(TopOpen ∘ 𝑅)) ∈ (TopOn‘(Base‘𝑌)) → (Base‘𝑌) = (∏t‘(TopOpen ∘ 𝑅)))
44 mpteq1 5118 . . . . . . . . 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 484 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝐼𝑊)
4737adantr 484 . . . . . . . . 9 ((𝜑𝑦𝐼) → (TopOpen ∘ 𝑅):𝐼⟶Top)
48 simpr 488 . . . . . . . . 9 ((𝜑𝑦𝐼) → 𝑦𝐼)
49 eqid 2738 . . . . . . . . . 10 (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅))
5049, 15ptpjcn 22362 . . . . . . . . 9 ((𝐼𝑊 ∧ (TopOpen ∘ 𝑅):𝐼⟶Top ∧ 𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5146, 47, 48, 50syl3anc 1372 . . . . . . . 8 ((𝜑𝑦𝐼) → (𝑥 (∏t‘(TopOpen ∘ 𝑅)) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5245, 51eqeltrd 2833 . . . . . . 7 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
5341, 27oveq12d 7188 . . . . . . 7 ((𝜑𝑦𝐼) → ((∏t‘(TopOpen ∘ 𝑅)) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
5452, 53eleqtrd 2835 . . . . . 6 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ (𝑥𝑦)) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
55 eqid 2738 . . . . . . . 8 (invg‘(𝑅𝑦)) = (invg‘(𝑅𝑦))
5629, 55tgpinv 22836 . . . . . . 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 22415 . . . . 5 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
5927oveq2d 7186 . . . . 5 ((𝜑𝑦𝐼) → ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)) = ((TopOpen‘𝑌) Cn (TopOpen‘(𝑅𝑦))))
6058, 59eleqtrrd 2836 . . . 4 ((𝜑𝑦𝐼) → (𝑥 ∈ (Base‘𝑌) ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))) ∈ ((TopOpen‘𝑌) Cn ((TopOpen ∘ 𝑅)‘𝑦)))
6115, 19, 2, 37, 60ptcn 22378 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))) ∈ ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
62 eqid 2738 . . . . . . 7 (invg𝑌) = (invg𝑌)
6317, 62grpinvf 18268 . . . . . 6 (𝑌 ∈ Grp → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
649, 63syl 17 . . . . 5 (𝜑 → (invg𝑌):(Base‘𝑌)⟶(Base‘𝑌))
6564feqmptd 6737 . . . 4 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)))
662adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝐼𝑊)
673adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑆𝑉)
688adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑅:𝐼⟶Grp)
69 simpr 488 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑌)) → 𝑥 ∈ (Base‘𝑌))
701, 66, 67, 68, 17, 62, 69prdsinvgd 18328 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑌)) → ((invg𝑌)‘𝑥) = (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦))))
7170mpteq2dva 5125 . . . 4 (𝜑 → (𝑥 ∈ (Base‘𝑌) ↦ ((invg𝑌)‘𝑥)) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
7265, 71eqtrd 2773 . . 3 (𝜑 → (invg𝑌) = (𝑥 ∈ (Base‘𝑌) ↦ (𝑦𝐼 ↦ ((invg‘(𝑅𝑦))‘(𝑥𝑦)))))
7339oveq2d 7186 . . 3 (𝜑 → ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)) = ((TopOpen‘𝑌) Cn (∏t‘(TopOpen ∘ 𝑅))))
7461, 72, 733eltr4d 2848 . 2 (𝜑 → (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌)))
7516, 62istgp 22828 . 2 (𝑌 ∈ TopGrp ↔ (𝑌 ∈ Grp ∧ 𝑌 ∈ TopMnd ∧ (invg𝑌) ∈ ((TopOpen‘𝑌) Cn (TopOpen‘𝑌))))
769, 14, 74, 75syl3anbrc 1344 1 (𝜑𝑌 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3053  Vcvv 3398  wss 3843   cuni 4796  cmpt 5110  ccom 5529   Fn wfn 6334  wf 6335  cfv 6339  (class class class)co 7170  Basecbs 16586  TopOpenctopn 16798  tcpt 16815  Xscprds 16822  Grpcgrp 18219  invgcminusg 18220  Topctop 21644  TopOnctopon 21661   Cn ccn 21975  TopMndctmd 22821  TopGrpctgp 22822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fi 8948  df-sup 8979  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-ip 16686  df-tset 16687  df-ple 16688  df-ds 16690  df-hom 16692  df-cco 16693  df-rest 16799  df-topn 16800  df-0g 16818  df-topgen 16820  df-pt 16821  df-prds 16824  df-plusf 17967  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-grp 18222  df-minusg 18223  df-top 21645  df-topon 21662  df-topsp 21684  df-bases 21697  df-cn 21978  df-cnp 21979  df-tx 22313  df-tmd 22823  df-tgp 22824
This theorem is referenced by: (None)
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