Theorem cntzfval 18926

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 6774	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4		cntzfval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
5	3, 4	eqtr4di 2796	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
6	5	pweqd 4552	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
7		fveq2 6774	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
8		cntzfval.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
9	7, 8	eqtr4di 2796	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = + )
10	9	oveqd 7292	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥 + 𝑦))
11	9	oveqd 7292	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑦(+g‘𝑚)𝑥) = (𝑦 + 𝑥))
12	10, 11	eqeq12d 2754	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13	12	ralbidv 3112	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥) ↔ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
14	5, 13	rabeqbidv 3420	. . . . 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 5165	. . . 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 18923	. . . 4 ⊢ Cntz = (𝑚 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑚) ↦ {𝑥 ∈ (Base‘𝑚) ∣ ∀𝑦 ∈ 𝑠 (𝑥(+g‘𝑚)𝑦) = (𝑦(+g‘𝑚)𝑥)}))
17	4	fvexi 6788	. . . . . 6 ⊢ 𝐵 ∈ V
18	17	pwex 5303	. . . . 5 ⊢ 𝒫 𝐵 ∈ V
19	18	mptex 7099	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}) ∈ V
20	15, 16, 19	fvmpt 6875	. . 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 2790	1 ⊢ (𝑀 ∈ 𝑉 → 𝑍 = (𝑠 ∈ 𝒫 𝐵 ↦ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝑠 (𝑥 + 𝑦) = (𝑦 + 𝑥)}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432  𝒫 cpw 4533   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Cntzccntz 18921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-cntz 18923

This theorem is referenced by:  cntzval  18927  cntzrcl  18933