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Theorem cntzi 19347
Description: Membership in a centralizer (inference). (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 22-Sep-2015.)
Hypotheses
Ref Expression
cntzi.p + = (+g𝑀)
cntzi.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntzi ((𝑋 ∈ (𝑍𝑆) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cntzi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
2 cntzi.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzrcl 19345 . . . . 5 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
4 cntzi.p . . . . . 6 + = (+g𝑀)
51, 4, 2elcntz 19340 . . . . 5 (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
63, 5simpl2im 503 . . . 4 (𝑋 ∈ (𝑍𝑆) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
76simplbda 499 . . 3 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑋 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
87anidms 566 . 2 (𝑋 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
9 oveq2 7439 . . . 4 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
10 oveq1 7438 . . . 4 (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
119, 10eqeq12d 2753 . . 3 (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
1211rspccva 3621 . 2 ((∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
138, 12sylan 580 1 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Cntzccntz 19333
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3062  df-rex 3071  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-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-id 5578  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-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-cntz 19335
This theorem is referenced by:  cntri  19350  cntzsgrpcl  19352  cntz2ss  19353  cntzsubm  19356  cntzsubg  19357  cntzmhm  19359  cntrsubgnsg  19361  lsmsubm  19671  lsmsubg  19672  lsmcom2  19673  subgdisj1  19709  subgdisj2  19710  pj1id  19717  pj1ghm  19721  gsumval3eu  19922  gsumval3  19925  gsumzaddlem  19939  gsumzoppg  19962  dprdfcntz  20035  cntzsubrng  20567  cntzsubr  20606  cntzsdrg  20803
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