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Theorem lsmsubg 19436
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 18950 . . . 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 18950 . . . 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 19435 . . 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 2736 . . . . . 6 (+g𝐺) = (+g𝐺)
1312, 8lsmelval 19431 . . . . 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 481 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16 subgrcl 18933 . . . . . . . . . 10 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝐺 ∈ Grp)
18 eqid 2736 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
1918subgss 18929 . . . . . . . . . . 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 3945 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (Base‘𝐺))
234adantr 481 . . . . . . . . . . 11 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
2418subgss 18929 . . . . . . . . . . 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 3945 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏 ∈ (Base‘𝐺))
28 eqid 2736 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
2918, 12, 28grpinvadd 18825 . . . . . . . . 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 481 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ⊆ (𝑍𝑈))
3228subginvcl 18937 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑇) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3315, 21, 32syl2anc 584 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3431, 33sseldd 3945 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ (𝑍𝑈))
3528subginvcl 18937 . . . . . . . . . 10 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏𝑈) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3623, 26, 35syl2anc 584 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3712, 9cntzi 19109 . . . . . . . . 9 ((((invg𝐺)‘𝑎) ∈ (𝑍𝑈) ∧ ((invg𝐺)‘𝑏) ∈ 𝑈) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3834, 36, 37syl2anc 584 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3930, 38eqtr4d 2779 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)))
4012, 8lsmelvali 19432 . . . . . . . 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 6842 . . . . . . 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 246 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4645rexlimdvva 3205 . . . 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 3142 . 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 18946 . . 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  SubMndcsubmnd 18600  Grpcgrp 18748  invgcminusg 18749  SubGrpcsubg 18922  Cntzccntz 19095  LSSumclsm 19416
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-subg 18925  df-cntz 19097  df-lsm 19418
This theorem is referenced by:  pj1ghm  19485  lsmsubg2  19637  dprd2da  19821  dmdprdsplit2lem  19824  dprdsplit  19827
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