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Theorem lsmsubg 19672
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 1137 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubGrp‘𝐺))
2 subgsubm 19166 . . . 4 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ∈ (SubMnd‘𝐺))
31, 2syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ∈ (SubMnd‘𝐺))
4 simp2 1138 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubGrp‘𝐺))
5 subgsubm 19166 . . . 4 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ∈ (SubMnd‘𝐺))
64, 5syl 17 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
7 simp3 1139 . . 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 19671 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
113, 6, 7, 10syl3anc 1373 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
12 eqid 2737 . . . . . 6 (+g𝐺) = (+g𝐺)
1312, 8lsmelval 19667 . . . . 5 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
14133adant3 1133 . . . 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 19149 . . . . . . . . . 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 19145 . . . . . . . . . . 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 771 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎𝑇)
2220, 21sseldd 3984 . . . . . . . . 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 19145 . . . . . . . . . . 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 773 . . . . . . . . . 10 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
2725, 26sseldd 3984 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏 ∈ (Base‘𝐺))
28 eqid 2737 . . . . . . . . . 10 (invg𝐺) = (invg𝐺)
2918, 12, 28grpinvadd 19036 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑎 ∈ (Base‘𝐺) ∧ 𝑏 ∈ (Base‘𝐺)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑏)(+g𝐺)((invg𝐺)‘𝑎)))
3017, 22, 27, 29syl3anc 1373 . . . . . . . 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 19153 . . . . . . . . . . 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 3984 . . . . . . . . 9 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘𝑎) ∈ (𝑍𝑈))
3528subginvcl 19153 . . . . . . . . . 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 19347 . . . . . . . . 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 2780 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) = (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)))
4012, 8lsmelvali 19668 . . . . . . . 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 839 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (((invg𝐺)‘𝑎)(+g𝐺)((invg𝐺)‘𝑏)) ∈ (𝑇 𝑈))
4239, 41eqeltrd 2841 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → ((invg𝐺)‘(𝑎(+g𝐺)𝑏)) ∈ (𝑇 𝑈))
43 fveq2 6906 . . . . . . 7 (𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) = ((invg𝐺)‘(𝑎(+g𝐺)𝑏)))
4443eleq1d 2826 . . . . . 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 3213 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4714, 46sylbid 240 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) → ((invg𝐺)‘𝑥) ∈ (𝑇 𝑈)))
4847ralrimiv 3145 . 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 19162 . . 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 713 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubGrp‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  SubMndcsubmnd 18795  Grpcgrp 18951  invgcminusg 18952  SubGrpcsubg 19138  Cntzccntz 19333  LSSumclsm 19652
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-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-en 8986  df-dom 8987  df-sdom 8988  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-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-subg 19141  df-cntz 19335  df-lsm 19654
This theorem is referenced by:  pj1ghm  19721  lsmsubg2  19877  dprd2da  20062  dmdprdsplit2lem  20065  dprdsplit  20068
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