Theorem cntzfval 19295

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 3450	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
3		fveq2 6840	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4		cntzfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
5	3, 4	eqtr4di 2789	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
6	5	pweqd 4558	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
8		cntzfval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
9	7, 8	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + )
10	9	oveqd 7384	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 7384	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2752	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	ralbidv 3160	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
14	5, 13	rabeqbidv 3407	. . . . 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 5172	. . . 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 19292	. . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
17	4	fvexi 6854	. . . . . 6 ⊢ 𝐵 ∈ V
18	17	pwex 5322	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
19	18	mptex 7178	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
20	15, 16, 19	fvmpt 6947	. . 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 2783	1 ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389  Vcvv 3429  𝒫 cpw 4541   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Cntzccntz 19290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-cntz 19292

This theorem is referenced by:  cntzval  19296  cntzrcl  19302