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Theorem lsmsubg 18190
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 1128 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2 subgsubm 17738 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
31, 2syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubMnd‘𝐺))
4 simp2 1129 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5 subgsubm 17738 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ∈ (SubMnd‘𝐺))
64, 5syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
7 simp3 1130 . . 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 18189 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
113, 6, 7, 10syl3anc 1439 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
12 eqid 2724 . . . . . 6 (+g𝐺) = (+g𝐺)
1312, 8lsmelval 18185 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
14133adant3 1124 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
151adantr 472 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16 subgrcl 17721 . . . . . . . . . 10 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝐺 ∈ Grp)
18 eqid 2724 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
1918subgss 17717 . . . . . . . . . . 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 811 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎𝑇)
2220, 21sseldd 3710 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (Base‘𝐺))
234adantr 472 . . . . . . . . . . 11 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
2418subgss 17717 . . . . . . . . . . 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 813 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
2725, 26sseldd 3710 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏 ∈ (Base‘𝐺))
28 eqid 2724 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
2918, 12, 28grpinvadd 17615 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3017, 22, 27, 29syl3anc 1439 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
317adantr 472 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ⊆ (𝑍𝑈))
3228subginvcl 17725 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑇) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3315, 21, 32syl2anc 696 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3431, 33sseldd 3710 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ (𝑍𝑈))
3528subginvcl 17725 . . . . . . . . . 10 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏𝑈) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3623, 26, 35syl2anc 696 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3712, 9cntzi 17883 . . . . . . . . 9 ((((invg𝐺)‘𝑎) ∈ (𝑍𝑈) ∧ ((invg𝐺)‘𝑏) ∈ 𝑈) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3834, 36, 37syl2anc 696 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3930, 38eqtr4d 2761 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)))
4012, 8lsmelvali 18186 . . . . . . . 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 1440 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) ∈ (𝑇 𝑈))
4239, 41eqeltrd 2803 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈))
43 fveq2 6304 . . . . . . 7 (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) = ((invg𝐺)‘(𝑎(+g𝐺)𝑏)))
4443eleq1d 2788 . . . . . 6 (𝑥 = (𝑎(+g𝐺)𝑏) → (((invg𝐺)‘𝑥) ∈ (𝑇 𝑈) ↔ ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈)))
4542, 44syl5ibrcom 237 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4645rexlimdvva 3140 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4714, 46sylbid 230 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4847ralrimiv 3067 . 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 17734 . . 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 995 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  wrex 3015  wss 3680  cfv 6001  (class class class)co 6765  Basecbs 15980  +gcplusg 16064  SubMndcsubmnd 17456  Grpcgrp 17544  invgcminusg 17545  SubGrpcsubg 17710  Cntzccntz 17869  LSSumclsm 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-0g 16225  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-grp 17547  df-minusg 17548  df-subg 17713  df-cntz 17871  df-lsm 18172
This theorem is referenced by:  pj1ghm  18237  lsmsubg2  18383  dprd2da  18562  dmdprdsplit2lem  18565  dprdsplit  18568
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