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Theorem cntzi 17969
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 2771 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
2 cntzi.z . . . . . . 7 𝑍 = (Cntz‘𝑀)
31, 2cntzrcl 17967 . . . . . 6 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
43simprd 483 . . . . 5 (𝑋 ∈ (𝑍𝑆) → 𝑆 ⊆ (Base‘𝑀))
5 cntzi.p . . . . . 6 + = (+g𝑀)
61, 5, 2elcntz 17962 . . . . 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 487 . . 3 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑋 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
98anidms 556 . 2 (𝑋 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
10 oveq2 6804 . . . 4 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
11 oveq1 6803 . . . 4 (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
1210, 11eqeq12d 2786 . . 3 (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
1312rspccva 3459 . 2 ((∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
149, 13sylan 569 1 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑌𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3723  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  Cntzccntz 17955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-cntz 17957
This theorem is referenced by:  cntri  17970  cntz2ss  17972  cntzsubm  17975  cntzsubg  17976  cntzmhm  17978  cntrsubgnsg  17980  lsmsubm  18275  lsmsubg  18276  lsmcom2  18277  subgdisj1  18311  subgdisj2  18312  pj1id  18319  pj1ghm  18323  gsumval3eu  18512  gsumval3  18515  gsumzaddlem  18528  gsumzoppg  18551  dprdfcntz  18622  cntzsubr  19022  cntzsdrg  38298
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