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Theorem lsmcom2 18782
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 3953 . . . . . . . 8 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ 𝑎𝑇) → 𝑎 ∈ (𝑍𝑈))
32adantrr 716 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑎 ∈ (𝑍𝑈))
4 simprr 772 . . . . . . 7 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → 𝑏𝑈)
5 eqid 2824 . . . . . . . 8 (+g𝐺) = (+g𝐺)
6 lsmsubg.z . . . . . . . 8 𝑍 = (Cntz‘𝐺)
75, 6cntzi 18461 . . . . . . 7 ((𝑎 ∈ (𝑍𝑈) ∧ 𝑏𝑈) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
83, 4, 7syl2anc 587 . . . . . 6 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑎(+g𝐺)𝑏) = (𝑏(+g𝐺)𝑎))
98eqeq2d 2835 . . . . 5 (((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑈)) → (𝑥 = (𝑎(+g𝐺)𝑏) ↔ 𝑥 = (𝑏(+g𝐺)𝑎)))
1092rexbidva 3291 . . . 4 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑎(+g𝐺)𝑏) ↔ ∃𝑎𝑇𝑏𝑈 𝑥 = (𝑏(+g𝐺)𝑎)))
11 rexcom 3346 . . . 4 (∃𝑎𝑇𝑏𝑈 𝑥 = (𝑏(+g𝐺)𝑎) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎))
1210, 11syl6bb 290 . . 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 18776 . . . 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 18776 . . . . 5 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ∈ (SubGrp‘𝐺)) → (𝑥 ∈ (𝑈 𝑇) ↔ ∃𝑏𝑈𝑎𝑇 𝑥 = (𝑏(+g𝐺)𝑎)))
1716ancoms 462 . . . 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 314 . 2 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ 𝑥 ∈ (𝑈 𝑇)))
2019eqrdv 2822 1 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wrex 3134  wss 3919  cfv 6345  (class class class)co 7151  +gcplusg 16567  SubGrpcsubg 18275  Cntzccntz 18447  LSSumclsm 18761
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687  df-subg 18278  df-cntz 18449  df-lsm 18763
This theorem is referenced by:  lsmdisj3  18811  lsmdisj3r  18814  lsmdisj3a  18817  lsmdisj3b  18818  pj2f  18826  pj1id  18827
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