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Theorem subgtgp 21819
Description: A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypothesis
Ref Expression
subgtgp.h 𝐻 = (𝐺s 𝑆)
Assertion
Ref Expression
subgtgp ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)

Proof of Theorem subgtgp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 subgtgp.h . . . 4 𝐻 = (𝐺s 𝑆)
21subggrp 17518 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantl 482 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4 tgptmd 21793 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5 subgsubm 17537 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
61submtmd 21818 . . 3 ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
74, 5, 6syl2an 494 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
81subgbas 17519 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
98adantl 482 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
109mpteq1d 4698 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
11 eqid 2621 . . . . . . . 8 (invg𝐺) = (invg𝐺)
12 eqid 2621 . . . . . . . 8 (invg𝐻) = (invg𝐻)
131, 11, 12subginv 17522 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1413adantll 749 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1514mpteq2dva 4704 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) = (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)))
16 eqid 2621 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
1716, 12grpinvf 17387 . . . . . . 7 (𝐻 ∈ Grp → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
183, 17syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
1918feqmptd 6206 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
2010, 15, 193eqtr4rd 2666 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)))
21 eqid 2621 . . . . 5 ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22 eqid 2621 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
23 eqid 2621 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2422, 23tgptopon 21796 . . . . . 6 (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2524adantr 481 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2623subgss 17516 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2726adantl 482 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28 tgpgrp 21792 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2928adantr 481 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3023, 11grpinvf 17387 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3129, 30syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3231feqmptd 6206 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
3322, 11tgpinv 21799 . . . . . . 7 (𝐺 ∈ TopGrp → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3433adantr 481 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3532, 34eqeltrrd 2699 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3621, 25, 27, 35cnmpt1res 21389 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3720, 36eqeltrd 2698 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
38 frn 6010 . . . . . 6 ((invg𝐻):(Base‘𝐻)⟶(Base‘𝐻) → ran (invg𝐻) ⊆ (Base‘𝐻))
3918, 38syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ (Base‘𝐻))
4039, 9sseqtr4d 3621 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ 𝑆)
41 cnrest2 21000 . . . 4 (((TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)) ∧ ran (invg𝐻) ⊆ 𝑆𝑆 ⊆ (Base‘𝐺)) → ((invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)) ↔ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
4225, 40, 27, 41syl3anc 1323 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)) ↔ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
4337, 42mpbid 222 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆)))
441, 22resstopn 20900 . . 3 ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
4544, 12istgp 21791 . 2 (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
463, 7, 43, 45syl3anbrc 1244 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555  cmpt 4673  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  s cress 15782  t crest 16002  TopOpenctopn 16003  SubMndcsubmnd 17255  Grpcgrp 17343  invgcminusg 17344  SubGrpcsubg 17509  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-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-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  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-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-tset 15881  df-rest 16004  df-topn 16005  df-0g 16023  df-topgen 16025  df-plusf 17162  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-grp 17346  df-minusg 17347  df-subg 17512  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cn 20941  df-tx 21275  df-tmd 21786  df-tgp 21787
This theorem is referenced by:  qqhcn  29817
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