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Theorem subgtgp 22705
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 18274 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantl 484 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4 tgptmd 22679 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5 subgsubm 18293 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
61submtmd 22704 . . 3 ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
74, 5, 6syl2an 597 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
81subgbas 18275 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
98adantl 484 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
109mpteq1d 5146 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
11 eqid 2819 . . . . . . . 8 (invg𝐺) = (invg𝐺)
12 eqid 2819 . . . . . . . 8 (invg𝐻) = (invg𝐻)
131, 11, 12subginv 18278 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1413adantll 712 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1514mpteq2dva 5152 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) = (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)))
16 eqid 2819 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
1716, 12grpinvf 18142 . . . . . . 7 (𝐻 ∈ Grp → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
183, 17syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
1918feqmptd 6726 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
2010, 15, 193eqtr4rd 2865 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)))
21 eqid 2819 . . . . 5 ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22 eqid 2819 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
23 eqid 2819 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2422, 23tgptopon 22682 . . . . . 6 (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2524adantr 483 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2623subgss 18272 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2726adantl 484 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28 tgpgrp 22678 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2928adantr 483 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3023, 11grpinvf 18142 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3129, 30syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3231feqmptd 6726 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
3322, 11tgpinv 22685 . . . . . . 7 (𝐺 ∈ TopGrp → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3433adantr 483 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3532, 34eqeltrrd 2912 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3621, 25, 27, 35cnmpt1res 22276 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3720, 36eqeltrd 2911 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3818frnd 6514 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ (Base‘𝐻))
3938, 9sseqtrrd 4006 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ 𝑆)
40 cnrest2 21886 . . . 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))))
4125, 39, 27, 40syl3anc 1365 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ((invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)) ↔ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
4237, 41mpbid 234 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆)))
431, 22resstopn 21786 . . 3 ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
4443, 12istgp 22677 . 2 (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
453, 7, 42, 44syl3anbrc 1337 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wss 3934  cmpt 5137  ran crn 5549  wf 6344  cfv 6348  (class class class)co 7148  Basecbs 16475  s cress 16476  t crest 16686  TopOpenctopn 16687  SubMndcsubmnd 17947  Grpcgrp 18095  invgcminusg 18096  SubGrpcsubg 18265  TopOnctopon 21510   Cn ccn 21824  TopMndctmd 22670  TopGrpctgp 22671
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 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fi 8867  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-tset 16576  df-rest 16688  df-topn 16689  df-0g 16707  df-topgen 16709  df-plusf 17843  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-subg 18268  df-top 21494  df-topon 21511  df-topsp 21533  df-bases 21546  df-cn 21827  df-tx 22162  df-tmd 22672  df-tgp 22673
This theorem is referenced by:  qqhcn  31220
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