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Theorem lsmsubg 18115
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 1081 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2 subgsubm 17663 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
31, 2syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubMnd‘𝐺))
4 simp2 1082 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5 subgsubm 17663 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ∈ (SubMnd‘𝐺))
64, 5syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
7 simp3 1083 . . 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 18114 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
113, 6, 7, 10syl3anc 1366 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
12 eqid 2651 . . . . . 6 (+g𝐺) = (+g𝐺)
1312, 8lsmelval 18110 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
14133adant3 1101 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
151adantr 480 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
16 subgrcl 17646 . . . . . . . . . 10 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1715, 16syl 17 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝐺 ∈ Grp)
18 eqid 2651 . . . . . . . . . . . 12 (Base‘𝐺) = (Base‘𝐺)
1918subgss 17642 . . . . . . . . . . 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 809 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎𝑇)
2220, 21sseldd 3637 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (Base‘𝐺))
234adantr 480 . . . . . . . . . . 11 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
2418subgss 17642 . . . . . . . . . . 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 811 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
2725, 26sseldd 3637 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏 ∈ (Base‘𝐺))
28 eqid 2651 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
2918, 12, 28grpinvadd 17540 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3017, 22, 27, 29syl3anc 1366 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
317adantr 480 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑇 ⊆ (𝑍𝑈))
3228subginvcl 17650 . . . . . . . . . . 11 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑎𝑇) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3315, 21, 32syl2anc 694 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ 𝑇)
3431, 33sseldd 3637 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ (𝑍𝑈))
3528subginvcl 17650 . . . . . . . . . 10 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑏𝑈) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3623, 26, 35syl2anc 694 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑏) ∈ 𝑈)
3712, 9cntzi 17808 . . . . . . . . 9 ((((invg𝐺)‘𝑎) ∈ (𝑍𝑈) ∧ ((invg𝐺)‘𝑏) ∈ 𝑈) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3834, 36, 37syl2anc 694 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3930, 38eqtr4d 2688 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)))
4012, 8lsmelvali 18111 . . . . . . . 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 1367 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) ∈ (𝑇 𝑈))
4239, 41eqeltrd 2730 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈))
43 fveq2 6229 . . . . . . 7 (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) = ((invg𝐺)‘(𝑎(+g𝐺)𝑏)))
4443eleq1d 2715 . . . . . 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 3067 . . . 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 2994 . 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 17659 . . 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 977 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 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  SubMndcsubmnd 17381  Grpcgrp 17469  invgcminusg 17470  SubGrpcsubg 17635  Cntzccntz 17794  LSSumclsm 18095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-subg 17638  df-cntz 17796  df-lsm 18097
This theorem is referenced by:  pj1ghm  18162  lsmsubg2  18308  dprd2da  18487  dmdprdsplit2lem  18490  dprdsplit  18493
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