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Theorem subgtgp 24053
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 19092 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantl 480 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4 tgptmd 24027 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5 subgsubm 19111 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
61submtmd 24052 . . 3 ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
74, 5, 6syl2an 594 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
81subgbas 19093 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
98adantl 480 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
109mpteq1d 5244 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
11 eqid 2725 . . . . . . . 8 (invg𝐺) = (invg𝐺)
12 eqid 2725 . . . . . . . 8 (invg𝐻) = (invg𝐻)
131, 11, 12subginv 19096 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1413adantll 712 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1514mpteq2dva 5249 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) = (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)))
16 eqid 2725 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
1716, 12grpinvf 18951 . . . . . . 7 (𝐻 ∈ Grp → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
183, 17syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
1918feqmptd 6966 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
2010, 15, 193eqtr4rd 2776 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)))
21 eqid 2725 . . . . 5 ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22 eqid 2725 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
23 eqid 2725 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2422, 23tgptopon 24030 . . . . . 6 (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2524adantr 479 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2623subgss 19090 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2726adantl 480 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28 tgpgrp 24026 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2928adantr 479 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3023, 11grpinvf 18951 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3129, 30syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3231feqmptd 6966 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
3322, 11tgpinv 24033 . . . . . . 7 (𝐺 ∈ TopGrp → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3433adantr 479 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3532, 34eqeltrrd 2826 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3621, 25, 27, 35cnmpt1res 23624 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3720, 36eqeltrd 2825 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3818frnd 6731 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ (Base‘𝐻))
3938, 9sseqtrrd 4018 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ 𝑆)
40 cnrest2 23234 . . . 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 1368 . . 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 231 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆)))
431, 22resstopn 23134 . . 3 ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
4443, 12istgp 24025 . 2 (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
453, 7, 42, 44syl3anbrc 1340 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wss 3944  cmpt 5232  ran crn 5679  wf 6545  cfv 6549  (class class class)co 7419  Basecbs 17183  s cress 17212  t crest 17405  TopOpenctopn 17406  SubMndcsubmnd 18742  Grpcgrp 18898  invgcminusg 18899  SubGrpcsubg 19083  TopOnctopon 22856   Cn ccn 23172  TopMndctmd 24018  TopGrpctgp 24019
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9436  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-tset 17255  df-rest 17407  df-topn 17408  df-0g 17426  df-topgen 17428  df-plusf 18602  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-subg 19086  df-top 22840  df-topon 22857  df-topsp 22879  df-bases 22893  df-cn 23175  df-tx 23510  df-tmd 24020  df-tgp 24021
This theorem is referenced by:  qqhcn  33723
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