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Theorem lsmcom2 19622
Description: Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
Hypotheses
Ref Expression
lsmsubg.p = (LSSum‘𝐺)
lsmsubg.z 𝑍 = (Cntz‘𝐺)
Assertion
Ref Expression
lsmcom2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))

Proof of Theorem lsmcom2
Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 1135 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑍𝑈))
21sselda 3976 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ 𝑎𝑇) → 𝑎 ∈ (𝑍𝑈))
32adantrr 715 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (𝑍𝑈))
4 simprr 771 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
5 eqid 2725 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmsubg.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
75, 6cntzi 19292 . . . . . . 7 ((𝑎 ∈ (𝑍𝑈) ∧ 𝑏𝑈) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
83, 4, 7syl2anc 582 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
98eqeq2d 2736 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑥 = (𝑎(+g𝐺)𝑏) ↔ 𝑥 = (𝑏(+g𝐺)𝑎)))
1092rexbidva 3207 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑏(+g𝐺)𝑎)))
11 rexcom 3277 . . . 4 (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑏(+g𝐺)𝑎) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎))
1210, 11bitrdi 286 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎)))
13 lsmsubg.p . . . . 5 = (LSSum‘𝐺)
145, 13lsmelval 19616 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
15143adant3 1129 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏)))
165, 13lsmelval 19616 . . . . 5 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎)))
1716ancoms 457 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎)))
18173adant3 1129 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎)))
1912, 15, 183bitr4d 310 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (𝑈 𝑇)))
2019eqrdv 2723 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3059  wss 3944  cfv 6549  (class class class)co 7419  +gcplusg 17236  SubGrpcsubg 19083  Cntzccntz 19278  LSSumclsm 19601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-subg 19086  df-cntz 19280  df-lsm 19603
This theorem is referenced by:  lsmdisj3  19650  lsmdisj3r  19653  lsmdisj3a  19656  lsmdisj3b  19657  pj2f  19665  pj1id  19666
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