Theorem cntzfval 19233

Description: First level substitution for a centralizer. (Contributed by Stefan O'Rear, 5-Sep-2015.)

Hypotheses

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

Assertion

Ref	Expression
cntzfval	⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Distinct variable groups:   𝑥,𝑠,𝑦, +   𝐵,𝑠,𝑥   𝑀,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑦)   𝑉(𝑥,𝑦,𝑠)   𝑍(𝑥,𝑦,𝑠)

Proof of Theorem cntzfval

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzfval.z	. 2 ⊢ 𝑍 = (Cntz‘𝑀)
2		elex 3487	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
3		fveq2 6884	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4		cntzfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
5	3, 4	eqtr4di 2784	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
6	5	pweqd 4614	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7		fveq2 6884	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
8		cntzfval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
9	7, 8	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + )
10	9	oveqd 7421	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 7421	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2742	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	ralbidv 3171	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
14	5, 13	rabeqbidv 3443	. . . . 5 ⊢ (𝑚 = 𝑀 → {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)})
15	6, 14	mpteq12dv 5232	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
16		df-cntz 19230	. . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
17	4	fvexi 6898	. . . . . 6 ⊢ 𝐵 ∈ V
18	17	pwex 5371	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
19	18	mptex 7219	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
20	15, 16, 19	fvmpt 6991	. . 3 ⊢ (𝑀 ∈ V → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
21	2, 20	syl 17	. 2 ⊢ (𝑀 ∈ 𝑉 → (Cntz‘𝑀) = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))
22	1, 21	eqtrid 2778	1 ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3055  {crab 3426  Vcvv 3468  𝒫 cpw 4597   ↦ cmpt 5224  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  +_gcplusg 17203  Cntzccntz 19228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-cntz 19230

This theorem is referenced by:  cntzval  19234  cntzrcl  19240