Theorem mzpclval 37288

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 6658	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 𝑣) = (ℤ ↑𝑚 𝑉))
2	1	oveq2d 6666	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
3	2	pweqd 4163	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑣)) = 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)))
4	1	xpeq1d 5138	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑𝑚 𝑣) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑎}))
5	4	eleq1d 2686	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 2986	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4187	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5139	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑𝑚 𝑉) × {𝑎}) = ((ℤ ↑𝑚 𝑉) × {𝑖}))
9	8	eleq1d 2686	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralv 3171	. . . . . 6 ⊢ (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝)
11	6, 10	syl6bb 276	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑𝑚 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑𝑚 𝑉) × {𝑖}) ∈ 𝑝))
12	1	mpteq1d 4738	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2686	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
14	13	raleqbi1dv 3146	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑𝑚 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
15		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 4745	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2686	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 4750	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2692	. . . . . . . 8 ⊢ ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
21	17, 20	syl6bb 276	. . . . . . 7 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
22	21	cbvralv 3171	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑𝑚 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
23	14, 22	syl6bb 276	. . . . 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 747	. . . 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 741	. . 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 3195	. 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 37286	. 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 6678	. . . 4 ⊢ (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
29	28	pwex 4848	. . 3 ⊢ 𝒫 (ℤ ↑𝑚 (ℤ ↑𝑚 𝑉)) ∈ V
30	29	rabex 4813	. 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 6282	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 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729   × cxp 5112  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857   + caddc 9939   · cmul 9941  ℤcz 11377  mzPolyCldcmzpcl 37284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-mzpcl 37286

This theorem is referenced by:  elmzpcl  37289