Theorem elcntz 18451

Description: Elementhood in the centralizer. (Contributed by Mario Carneiro, 22-Sep-2015.)

Hypotheses

Ref	Expression
cntzfval.b	⊢ 𝐵 = (Base‘𝑀)
cntzfval.p	⊢ + = (+g‘𝑀)
cntzfval.z	⊢ 𝑍 = (Cntz‘𝑀)

Assertion

Ref	Expression
elcntz	⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))

Distinct variable groups:   𝑦, +   𝑦,𝐴   𝑦,𝑀   𝑦,𝑆

Allowed substitution hints:   𝐵(𝑦)   𝑍(𝑦)

Proof of Theorem elcntz

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
2		cntzfval.p	. . . 4 ⊢ + = (+g‘𝑀)
3		cntzfval.z	. . . 4 ⊢ 𝑍 = (Cntz‘𝑀)
4	1, 2, 3	cntzval 18450	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
5	4	eleq2d 2898	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6		oveq1 7162	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7		oveq2 7163	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
8	6, 7	eqeq12d 2837	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
9	8	ralbidv 3197	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
10	9	elrab 3679	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
11	5, 10	syl6bb 289	1 ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142   ⊆ wss 3935  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  +_gcplusg 16564  Cntzccntz 18444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-cntz 18446

This theorem is referenced by:  cntzel  18452  cntzi  18458  resscntz  18461  cntzsubm  18465  cntzmhm  18468  oppgcntz  18491  dprdfcntz  19136  cntzun  30695