Theorem mzpclval 42698

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 7361	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
2	1	oveq2d 7369	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
3	2	pweqd 4570	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
4	1	xpeq1d 5652	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
5	4	eleq1d 2813	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 3152	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4589	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5653	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
9	8	eleq1d 2813	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralvw 3207	. . . . . 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 5185	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2813	. . . . . . 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 6826	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 5189	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2813	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6825	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 5199	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2819	. . . . . . . 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 3207	. . . . . 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 3415	. 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 42696	. 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 7386	. . . 4 ⊢ (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
29	28	pwex 5322	. . 3 ⊢ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
30	29	rabex 5281	. 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 6934	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 1540   ∈ wcel 2109  ∀wral 3044  {crab 3396  Vcvv 3438  𝒫 cpw 4553  {csn 4579   ↦ cmpt 5176   × cxp 5621  ‘cfv 6486  (class class class)co 7353   ∘_f cof 7615   ↑_m cmap 8760   + caddc 11031   · cmul 11033  ℤcz 12489  mzPolyCldcmzpcl 42694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-mzpcl 42696

This theorem is referenced by:  elmzpcl  42699