Theorem elcntz 19563

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 19562	. . 3 ⊢ (𝑆 ⊆ 𝐵 → (𝑍‘𝑆) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
5	4	eleq2d 2850	. 2 ⊢ (𝑆 ⊆ 𝐵 → (𝐴 ∈ (𝑍‘𝑆) ↔ 𝐴 ∈ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
6		oveq1 7447	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 + 𝑦) = (𝐴 + 𝑦))
7		oveq2 7448	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 + 𝑥) = (𝑦 + 𝐴))
8	6, 7	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝐴 + 𝑦) = (𝑦 + 𝐴)))
9	8	ralbidv 3189	. . 3 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ 𝑆 (𝐴 + 𝑦) = (𝑦 + 𝐴)))
10	9	elrab 3648	. 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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  Cntzccntz 19556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-cntz 19558

This theorem is used by:  cntzel  19564  cntzi  19570  elcntr  19571  resscntz  19574  cntzsgrpcl  19575  cntzsubm  19579  cntzmhm  19582  oppgcntz  19605  dprdfcntz  20258  rng2idl1cntr  21628  cntzun  33707