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Theorem subgtgp 24113
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 19147 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
32adantl 481 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ Grp)
4 tgptmd 24087 . . 3 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
5 subgsubm 19166 . . 3 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubMnd‘𝐺))
61submtmd 24112 . . 3 ((𝐺 ∈ TopMnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) → 𝐻 ∈ TopMnd)
74, 5, 6syl2an 596 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopMnd)
81subgbas 19148 . . . . . . 7 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
98adantl 481 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 = (Base‘𝐻))
109mpteq1d 5237 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
11 eqid 2737 . . . . . . . 8 (invg𝐺) = (invg𝐺)
12 eqid 2737 . . . . . . . 8 (invg𝐻) = (invg𝐻)
131, 11, 12subginv 19151 . . . . . . 7 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1413adantll 714 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) ∧ 𝑥𝑆) → ((invg𝐺)‘𝑥) = ((invg𝐻)‘𝑥))
1514mpteq2dva 5242 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) = (𝑥𝑆 ↦ ((invg𝐻)‘𝑥)))
16 eqid 2737 . . . . . . . 8 (Base‘𝐻) = (Base‘𝐻)
1716, 12grpinvf 19004 . . . . . . 7 (𝐻 ∈ Grp → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
183, 17syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻):(Base‘𝐻)⟶(Base‘𝐻))
1918feqmptd 6977 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥 ∈ (Base‘𝐻) ↦ ((invg𝐻)‘𝑥)))
2010, 15, 193eqtr4rd 2788 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) = (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)))
21 eqid 2737 . . . . 5 ((TopOpen‘𝐺) ↾t 𝑆) = ((TopOpen‘𝐺) ↾t 𝑆)
22 eqid 2737 . . . . . . 7 (TopOpen‘𝐺) = (TopOpen‘𝐺)
23 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
2422, 23tgptopon 24090 . . . . . 6 (𝐺 ∈ TopGrp → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2524adantr 480 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (TopOpen‘𝐺) ∈ (TopOn‘(Base‘𝐺)))
2623subgss 19145 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
2726adantl 481 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝑆 ⊆ (Base‘𝐺))
28 tgpgrp 24086 . . . . . . . . 9 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
2928adantr 480 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐺 ∈ Grp)
3023, 11grpinvf 19004 . . . . . . . 8 (𝐺 ∈ Grp → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3129, 30syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺):(Base‘𝐺)⟶(Base‘𝐺))
3231feqmptd 6977 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) = (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)))
3322, 11tgpinv 24093 . . . . . . 7 (𝐺 ∈ TopGrp → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3433adantr 480 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3532, 34eqeltrrd 2842 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (Base‘𝐺) ↦ ((invg𝐺)‘𝑥)) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺)))
3621, 25, 27, 35cnmpt1res 23684 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (𝑥𝑆 ↦ ((invg𝐺)‘𝑥)) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3720, 36eqeltrd 2841 . . 3 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn (TopOpen‘𝐺)))
3818frnd 6744 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ (Base‘𝐻))
3938, 9sseqtrrd 4021 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → ran (invg𝐻) ⊆ 𝑆)
40 cnrest2 23294 . . . 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 1373 . . 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 232 . 2 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆)))
431, 22resstopn 23194 . . 3 ((TopOpen‘𝐺) ↾t 𝑆) = (TopOpen‘𝐻)
4443, 12istgp 24085 . 2 (𝐻 ∈ TopGrp ↔ (𝐻 ∈ Grp ∧ 𝐻 ∈ TopMnd ∧ (invg𝐻) ∈ (((TopOpen‘𝐺) ↾t 𝑆) Cn ((TopOpen‘𝐺) ↾t 𝑆))))
453, 7, 42, 44syl3anbrc 1344 1 ((𝐺 ∈ TopGrp ∧ 𝑆 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ TopGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  cmpt 5225  ran crn 5686  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  t crest 17465  TopOpenctopn 17466  SubMndcsubmnd 18795  Grpcgrp 18951  invgcminusg 18952  SubGrpcsubg 19138  TopOnctopon 22916   Cn ccn 23232  TopMndctmd 24078  TopGrpctgp 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-tset 17316  df-rest 17467  df-topn 17468  df-0g 17486  df-topgen 17488  df-plusf 18652  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-subg 19141  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-tx 23570  df-tmd 24080  df-tgp 24081
This theorem is referenced by:  qqhcn  33992
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