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Theorem lsmsubg 19355
Description: The sum of two commuting subgroups is a subgroup. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmsubg.p = (LSSum‘𝐺)
lsmsubg.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
lsmsubg ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))

Proof of Theorem lsmsubg
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1136 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2 subgsubm 18873 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
31, 2syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubMnd‘𝐺))
4 simp2 1137 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5 subgsubm 18873 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ∈ (SubMnd‘𝐺))
64, 5syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
7 simp3 1138 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑍𝑈))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
9 lsmsubg.z . . . 4 𝑍 = (Cntz‘𝐺)
108, 9lsmsubm 19354 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
113, 6, 7, 10syl3anc 1371 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
12 eqid 2737 . . . . . 6 (+g𝐺) = (+g𝐺)
1312, 8lsmelval 19350 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
14133adant3 1132 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
151adantr 482 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16 subgrcl 18856 . . . . . . . . . 10 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝐺 ∈ Grp)
18 eqid 2737 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
1918subgss 18852 . . . . . . . . . . 11 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
2015, 19syl 17 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ⊆ (Base‘𝐺))
21 simprl 769 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎𝑇)
2220, 21sseldd 3936 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (Base‘𝐺))
234adantr 482 . . . . . . . . . . 11 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
2418subgss 18852 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
2523, 24syl 17 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑈 ⊆ (Base‘𝐺))
26 simprr 771 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
2725, 26sseldd 3936 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏 ∈ (Base‘𝐺))
28 eqid 2737 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
2918, 12, 28grpinvadd 18749 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3017, 22, 27, 29syl3anc 1371 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
317adantr 482 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ⊆ (𝑍𝑈))
3228subginvcl 18860 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑇) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3315, 21, 32syl2anc 585 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3431, 33sseldd 3936 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ (𝑍𝑈))
3528subginvcl 18860 . . . . . . . . . 10 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏𝑈) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3623, 26, 35syl2anc 585 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3712, 9cntzi 19031 . . . . . . . . 9 ((((invg𝐺)‘𝑎) ∈ (𝑍𝑈) ∧ ((invg𝐺)‘𝑏) ∈ 𝑈) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3834, 36, 37syl2anc 585 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3930, 38eqtr4d 2780 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)))
4012, 8lsmelvali 19351 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) ∧ (((invg𝐺)‘𝑎) ∈ 𝑇 ∧ ((invg𝐺)‘𝑏) ∈ 𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) ∈ (𝑇 𝑈))
4115, 23, 33, 36, 40syl22anc 837 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) ∈ (𝑇 𝑈))
4239, 41eqeltrd 2838 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈))
43 fveq2 6829 . . . . . . 7 (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) = ((invg𝐺)‘(𝑎(+g𝐺)𝑏)))
4443eleq1d 2822 . . . . . 6 (𝑥 = (𝑎(+g𝐺)𝑏) → (((invg𝐺)‘𝑥) ∈ (𝑇 𝑈) ↔ ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈)))
4542, 44syl5ibrcom 247 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4645rexlimdvva 3202 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4714, 46sylbid 239 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4847ralrimiv 3139 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)((invg𝐺)‘𝑥) ∈ (𝑇 𝑈))
491, 16syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Grp)
5028issubg3 18869 . . 3 (𝐺 ∈ Grp → ((𝑇 𝑈) ∈ (SubGrp‘𝐺) ↔ ((𝑇 𝑈) ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ (𝑇 𝑈)((invg𝐺)‘𝑥) ∈ (𝑇 𝑈))))
5149, 50syl 17 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑇 𝑈) ∈ (SubGrp‘𝐺) ↔ ((𝑇 𝑈) ∈ (SubMnd‘𝐺) ∧ ∀𝑥 ∈ (𝑇 𝑈)((invg𝐺)‘𝑥) ∈ (𝑇 𝑈))))
5211, 48, 51mpbir2and 711 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071  wss 3901  cfv 6483  (class class class)co 7341  Basecbs 17009  +gcplusg 17059  SubMndcsubmnd 18526  Grpcgrp 18673  invgcminusg 18674  SubGrpcsubg 18845  Cntzccntz 19017  LSSumclsm 19335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-er 8573  df-en 8809  df-dom 8810  df-sdom 8811  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-2 12141  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-grp 18676  df-minusg 18677  df-subg 18848  df-cntz 19019  df-lsm 19337
This theorem is referenced by:  pj1ghm  19404  lsmsubg2  19555  dprd2da  19739  dmdprdsplit2lem  19742  dprdsplit  19745
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