Theorem mzpclval 42737

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 7349	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
2	1	oveq2d 7357	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
3	2	pweqd 4565	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
4	1	xpeq1d 5643	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
5	4	eleq1d 2814	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 3153	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4584	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5644	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
9	8	eleq1d 2814	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralvw 3208	. . . . . 6 ⊢ (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝)
11	6, 10	bitrdi 287	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
12	1	mpteq1d 5179	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2814	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
14	13	raleqbi1dv 3302	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
15		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 5183	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2814	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6816	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 5193	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2820	. . . . . . . 8 ⊢ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
21	17, 20	bitrdi 287	. . . . . . 7 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
22	21	cbvralvw 3208	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
23	14, 22	bitrdi 287	. . . . 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 632	. . . 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 631	. . 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 3411	. 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 42735	. 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 7374	. . . 4 ⊢ (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
29	28	pwex 5316	. . 3 ⊢ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
30	29	rabex 5275	. 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 6924	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 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  {crab 3393  Vcvv 3434  𝒫 cpw 4548  {csn 4574   ↦ cmpt 5170   × cxp 5612  ‘cfv 6477  (class class class)co 7341   ∘_f cof 7603   ↑_m cmap 8745   + caddc 11001   · cmul 11003  ℤcz 12460  mzPolyCldcmzpcl 42733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-mzpcl 42735

This theorem is referenced by:  elmzpcl  42738