Theorem elcntzsn 19204

Description: Value of the centralizer of a singleton. (Contributed by Mario Carneiro, 25-Apr-2016.)

Hypotheses

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

Assertion

Ref	Expression
elcntzsn	⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))

Proof of Theorem elcntzsn

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	cntzsnval 19203	. . 3 ⊢ (𝑌 ∈ 𝐵 → (𝑍‘{𝑌}) = {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)})
5	4	eleq2d 2814	. 2 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)}))
6		oveq1 7356	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 + 𝑌) = (𝑋 + 𝑌))
7		oveq2 7357	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑌 + 𝑥) = (𝑌 + 𝑋))
8	6, 7	eqeq12d 2745	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 + 𝑌) = (𝑌 + 𝑥) ↔ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
9	8	elrab 3648	. 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ (𝑥 + 𝑌) = (𝑌 + 𝑥)} ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋)))
10	5, 9	bitrdi 287	1 ⊢ (𝑌 ∈ 𝐵 → (𝑋 ∈ (𝑍‘{𝑌}) ↔ (𝑋 ∈ 𝐵 ∧ (𝑋 + 𝑌) = (𝑌 + 𝑋))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394  {csn 4577  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  +_gcplusg 17161  Cntzccntz 19194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-cntz 19196

This theorem is referenced by:  gsumconst  19813  gsumpt  19841  cntzsnid  33031