Theorem cntzi 19543

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	⊢ + = (+g‘𝑀)
cntzi.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
cntzi	⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cntzi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		cntzi.z	. . . . . 6 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzrcl 19541	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
4		cntzi.p	. . . . . 6 ⊢ + = (+g‘𝑀)
5	1, 4, 2	elcntz 19536	. . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
6	3, 5	simpl2im 513	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
7	6	simplbda 505	. . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
8	7	anidms 577	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
9		oveq2 7428	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
10		oveq1 7427	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
11	9, 10	eqeq12d 2776	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
12	11	rspccva 3575	. 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
13	8, 12	sylan 592	1 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Cntzccntz 19529

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-cntz 19531

This theorem is used by:  cntri  19546  cntzsgrpcl  19548  cntz2ss  19549  cntzsubm  19552  cntzsubg  19553  cntzmhm  19555  cntrsubgnsg  19557  lsmsubm  19867  lsmsubg  19868  lsmcom2  19869  subgdisj1  19905  subgdisj2  19906  pj1id  19913  pj1ghm  19917  gsumval3eu  20118  gsumval3  20121  gsumzaddlem  20135  gsumzoppg  20158  dprdfcntz  20231  cntzsubrng  20819  cntzsubr  20858  cntzsdrg  21059