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Theorem subgtgp 22131
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 17819 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantl 473 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4 tgptmd 22105 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5 subgsubm 17838 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
61submtmd 22130 . . 3 ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
74, 5, 6syl2an 495 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
81subgbas 17820 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
98adantl 473 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
109mpteq1d 4891 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
11 eqid 2761 . . . . . . . 8 (invg𝐺) = (invg𝐺)
12 eqid 2761 . . . . . . . 8 (invg𝐻) = (invg𝐻)
131, 11, 12subginv 17823 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1413adantll 752 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1514mpteq2dva 4897 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) = (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)))
16 eqid 2761 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
1716, 12grpinvf 17688 . . . . . . 7 (𝐻 ∈ Grp → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
183, 17syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
1918feqmptd 6413 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
2010, 15, 193eqtr4rd 2806 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)))
21 eqid 2761 . . . . 5 ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22 eqid 2761 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
23 eqid 2761 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2422, 23tgptopon 22108 . . . . . 6 (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2524adantr 472 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2623subgss 17817 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2726adantl 473 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28 tgpgrp 22104 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2928adantr 472 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3023, 11grpinvf 17688 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3129, 30syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3231feqmptd 6413 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
3322, 11tgpinv 22111 . . . . . . 7 (𝐺 ∈ TopGrp → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3433adantr 472 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3532, 34eqeltrrd 2841 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3621, 25, 27, 35cnmpt1res 21702 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3720, 36eqeltrd 2840 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
38 frn 6215 . . . . . 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 3784 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ 𝑆)
41 cnrest2 21313 . . . 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 1477 . . 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 21213 . . 3 ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
4544, 12istgp 22103 . 2 (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
463, 7, 43, 45syl3anbrc 1429 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wss 3716  cmpt 4882  ran crn 5268  wf 6046  cfv 6050  (class class class)co 6815  Basecbs 16080  s cress 16081  t crest 16304  TopOpenctopn 16305  SubMndcsubmnd 17556  Grpcgrp 17644  invgcminusg 17645  SubGrpcsubg 17810  TopOnctopon 20938   Cn ccn 21251  TopMndctmd 22096  TopGrpctgp 22097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-oadd 7735  df-er 7914  df-map 8028  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-fi 8485  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-3 11293  df-4 11294  df-5 11295  df-6 11296  df-7 11297  df-8 11298  df-9 11299  df-ndx 16083  df-slot 16084  df-base 16086  df-sets 16087  df-ress 16088  df-plusg 16177  df-tset 16183  df-rest 16306  df-topn 16307  df-0g 16325  df-topgen 16327  df-plusf 17463  df-mgm 17464  df-sgrp 17506  df-mnd 17517  df-submnd 17558  df-grp 17647  df-minusg 17648  df-subg 17813  df-top 20922  df-topon 20939  df-topsp 20960  df-bases 20973  df-cn 21254  df-tx 21588  df-tmd 22098  df-tgp 22099
This theorem is referenced by:  qqhcn  30366
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