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Theorem cntzi 18397
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 2818 . . . . . 6 (Base‘𝑀) = (Base‘𝑀)
2 cntzi.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzrcl 18395 . . . . 5 (𝑋 ∈ (𝑍𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
4 cntzi.p . . . . . 6 + = (+g𝑀)
51, 4, 2elcntz 18390 . . . . 5 (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
63, 5simpl2im 504 . . . 4 (𝑋 ∈ (𝑍𝑆) → (𝑋 ∈ (𝑍𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
76simplbda 500 . . 3 ((𝑋 ∈ (𝑍𝑆) ∧ 𝑋 ∈ (𝑍𝑆)) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
87anidms 567 . 2 (𝑋 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
9 oveq2 7153 . . . 4 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
10 oveq1 7152 . . . 4 (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
119, 10eqeq12d 2834 . . 3 (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
1211rspccva 3619 . 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-cntz 18385
This theorem is referenced by:  cntri  18399  cntz2ss  18401  cntzsubm  18404  cntzsubg  18405  cntzmhm  18407  cntrsubgnsg  18409  lsmsubm  18707  lsmsubg  18708  lsmcom2  18709  subgdisj1  18746  subgdisj2  18747  pj1id  18754  pj1ghm  18758  gsumval3eu  18953  gsumval3  18956  gsumzaddlem  18970  gsumzoppg  18993  dprdfcntz  19066  cntzsubr  19497  cntzsdrg  19510
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