Theorem elcntz 19395

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 19394	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
5	4	eleq2d 2856	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6		oveq1 7421	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7		oveq2 7422	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
8	6, 7	eqeq12d 2786	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
9	8	ralbidv 3195	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
10	9	elrab 3658	. 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)} ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
11	5, 10	bitrdi 290	1 ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  {crab 3423   ⊆ wss 3913  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  Cntzccntz 19388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-cntz 19390

This theorem is referenced by:  cntzel  19396  cntzi  19402  elcntr  19403  resscntz  19406  cntzsgrpcl  19407  cntzsubm  19411  cntzmhm  19414  oppgcntz  19437  dprdfcntz  20090  rng2idl1cntr  21428  cntzun  33369