Theorem mzpclval 36306

Description: Substitution lemma for mzPolyCld. (Contributed by Stefan O'Rear, 4-Oct-2014.)

Assertion

Ref	Expression
mzpclval	⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))})

Distinct variable groups:   𝑉,𝑝,𝑓,𝑔   𝑖,𝑉,𝑝   𝑗,𝑉,𝑥,𝑝

Proof of Theorem mzpclval

Dummy variables 𝑣 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6557	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
2	1	oveq2d 6565	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
3	2	pweqd 4113	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
4	1	xpeq1d 5062	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑𝑚 𝑣) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑎}))
5	4	eleq1d 2672	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 2969	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4135	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5063	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑𝑚 𝑉) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑖}))
9	8	eleq1d 2672	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralv 3147	. . . . . 6 ⊢ (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝)
11	6, 10	syl6bb 275	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
12	1	mpteq1d 4666	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2672	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
14	13	raleqbi1dv 3123	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
15		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 4673	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2672	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6102	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 4678	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2679	. . . . . . . 8 ⊢ ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
21	17, 20	syl6bb 275	. . . . . . 7 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
22	21	cbvralv 3147	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
23	14, 22	syl6bb 275	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
24	11, 23	anbi12d 743	. . . 4 ⊢ (𝑣 = 𝑉 → ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝) ↔ (∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)))
25	24	anbi1d 737	. . 3 ⊢ (𝑣 = 𝑉 → (((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝)) ↔ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))))
26	3, 25	rabeqbidv 3168	. 2 ⊢ (𝑣 = 𝑉 → {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∣ ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))} = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))})
27		df-mzpcl 36304	. 2 ⊢ mzPolyCld = (𝑣 ∈ V ↦ {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) ∣ ((∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ∧ ∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))})
28		ovex 6577	. . . 4 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
29	28	pwex 4774	. . 3 ⊢ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
30	29	rabex 4740	. 2 ⊢ {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))} ∈ V
31	26, 27, 30	fvmpt 6191	1 ⊢ (𝑉 ∈ V → (mzPolyCld‘𝑉) = {𝑝 ∈ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∣ ((∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝 ∧ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝) ∧ ∀𝑓 ∈ 𝑝 ∀𝑔 ∈ 𝑝 ((𝑓 ∘𝑓 + 𝑔) ∈ 𝑝 ∧ (𝑓 ∘𝑓 · 𝑔) ∈ 𝑝))})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793   ↑_𝑚 cmap 7744   + caddc 9818   · cmul 9820  ℤcz 11254  mzPolyCldcmzpcl 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-mzpcl 36304

This theorem is referenced by:  elmzpcl  36307