Theorem cntzi 17762

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 2622	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		cntzi.z	. . . . . . 7 ⊢ 𝑍 = (Cntz‘𝑀)
3	1, 2	cntzrcl 17760	. . . . . 6 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑀 ∈ V ∧ 𝑆 ⊆ (Base‘𝑀)))
4	3	simprd 479	. . . . 5 ⊢ (𝑋 ∈ (𝑍‘𝑆) → 𝑆 ⊆ (Base‘𝑀))
5		cntzi.p	. . . . . 6 ⊢ + = (+g‘𝑀)
6	1, 5, 2	elcntz 17755	. . . . 5 ⊢ (𝑆 ⊆ (Base‘𝑀) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
7	4, 6	syl 17	. . . 4 ⊢ (𝑋 ∈ (𝑍‘𝑆) → (𝑋 ∈ (𝑍‘𝑆) ↔ (𝑋 ∈ (Base‘𝑀) ∧ ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))))
8	7	simplbda 654	. . 3 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑋 ∈ (𝑍‘𝑆)) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
9	8	anidms 677	. 2 ⊢ (𝑋 ∈ (𝑍‘𝑆) → ∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋))
10		oveq2 6658	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
11		oveq1 6657	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 + 𝑋) = (𝑌 + 𝑋))
12	10, 11	eqeq12d 2637	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 + 𝑦) = (𝑦 + 𝑋) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
13	12	rspccva 3308	. 2 ⊢ ((∀𝑦 ∈ 𝑆 (𝑋 + 𝑦) = (𝑦 + 𝑋) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
14	9, 13	sylan 488	1 ⊢ ((𝑋 ∈ (𝑍‘𝑆) ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  Cntzccntz 17748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-cntz 17750

This theorem is referenced by:  cntri  17763  cntz2ss  17765  cntzsubm  17768  cntzsubg  17769  cntzmhm  17771  cntrsubgnsg  17773  lsmsubm  18068  lsmsubg  18069  lsmcom2  18070  subgdisj1  18104  subgdisj2  18105  pj1id  18112  pj1ghm  18116  gsumval3eu  18305  gsumval3  18308  gsumzaddlem  18321  gsumzoppg  18344  dprdfcntz  18414  cntzsubr  18812  cntzsdrg  37772