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Theorem cntzi 17743
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 2620 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
2 cntzi.z . . . . . . 7 𝑍 = (Cntz‘𝑀)
31, 2cntzrcl 17741 . . . . . 6 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
43simprd 479 . . . . 5 (𝑋 ∈ (𝑍𝑆) → 𝑆 ⊆ (Base‘𝑀))
5 cntzi.p . . . . . 6 + = (+g𝑀)
61, 5, 2elcntz 17736 . . . . 5 (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
74, 6syl 17 . . . 4 (𝑋 ∈ (𝑍𝑆) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
87simplbda 653 . . 3 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑋 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
98anidms 676 . 2 (𝑋 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
10 oveq2 6643 . . . 4 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
11 oveq1 6642 . . . 4 (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
1210, 11eqeq12d 2635 . . 3 (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
1312rspccva 3303 . 2 ((∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
149, 13sylan 488 1 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  Vcvv 3195  wss 3567  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Cntzccntz 17729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-cntz 17731
This theorem is referenced by:  cntri  17744  cntz2ss  17746  cntzsubm  17749  cntzsubg  17750  cntzmhm  17752  cntrsubgnsg  17754  lsmsubm  18049  lsmsubg  18050  lsmcom2  18051  subgdisj1  18085  subgdisj2  18086  pj1id  18093  pj1ghm  18097  gsumval3eu  18286  gsumval3  18289  gsumzaddlem  18302  gsumzoppg  18325  dprdfcntz  18395  cntzsubr  18793  cntzsdrg  37591
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