Theorem mzpclval 38662

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 6978	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
2	1	oveq2d 6986	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
3	2	pweqd 4421	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
4	1	xpeq1d 5429	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑𝑚 𝑣) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑎}))
5	4	eleq1d 2844	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 3141	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4445	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5430	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑𝑚 𝑉) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑖}))
9	8	eleq1d 2844	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralv 3377	. . . . . 6 ⊢ (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝)
11	6, 10	syl6bb 279	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
12	1	mpteq1d 5010	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2844	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
14	13	raleqbi1dv 3337	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
15		fveq2 6493	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 5017	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2844	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6492	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 5022	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2850	. . . . . . . 8 ⊢ ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
21	17, 20	syl6bb 279	. . . . . . 7 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
22	21	cbvralv 3377	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
23	14, 22	syl6bb 279	. . . . 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 621	. . . 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 620	. . 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 3402	. 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 38660	. 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 7002	. . . 4 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
29	28	pwex 5128	. . 3 ⊢ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
30	29	rabex 5085	. 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 6589	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 387   = wceq 1507   ∈ wcel 2048  ∀wral 3082  {crab 3086  Vcvv 3409  𝒫 cpw 4416  {csn 4435   ↦ cmpt 5002   × cxp 5398  ‘cfv 6182  (class class class)co 6970   ∘_𝑓 cof 7219   ↑_𝑚 cmap 8198   + caddc 10330   · cmul 10332  ℤcz 11786  mzPolyCldcmzpcl 38658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973  df-mzpcl 38660

This theorem is referenced by:  elmzpcl  38663