Theorem mzpclval 43426

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 7418	. . . . 5 ⊢ (𝑣 = 𝑉 → (ℤ ↑m 𝑣) = (ℤ ↑m 𝑉))
2	1	oveq2d 7426	. . . 4 ⊢ (𝑣 = 𝑉 → (ℤ ↑m (ℤ ↑m 𝑣)) = (ℤ ↑m (ℤ ↑m 𝑉)))
3	2	pweqd 4578	. . 3 ⊢ (𝑣 = 𝑉 → 𝒫 (ℤ ↑m (ℤ ↑m 𝑣)) = 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)))
4	1	xpeq1d 5690	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((ℤ ↑m 𝑣) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑎}))
5	4	eleq1d 2846	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
6	5	ralbidv 3186	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝))
7		sneq 4598	. . . . . . . . 9 ⊢ (𝑎 = 𝑖 → {𝑎} = {𝑖})
8	7	xpeq2d 5691	. . . . . . . 8 ⊢ (𝑎 = 𝑖 → ((ℤ ↑m 𝑉) × {𝑎}) = ((ℤ ↑m 𝑉) × {𝑖}))
9	8	eleq1d 2846	. . . . . . 7 ⊢ (𝑎 = 𝑖 → (((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
10	9	cbvralvw 3241	. . . . . 6 ⊢ (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝)
11	6, 10	bitrdi 290	. . . . 5 ⊢ (𝑣 = 𝑉 → (∀𝑎 ∈ ℤ ((ℤ ↑m 𝑣) × {𝑎}) ∈ 𝑝 ↔ ∀𝑖 ∈ ℤ ((ℤ ↑m 𝑉) × {𝑖}) ∈ 𝑝))
12	1	mpteq1d 5200	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)))
13	12	eleq1d 2846	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
14	13	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑣 = 𝑉 → (∀𝑏 ∈ 𝑣 (𝑐 ∈ (ℤ ↑m 𝑣) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝))
15		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑏 = 𝑗 → (𝑐‘𝑏) = (𝑐‘𝑗))
16	15	mpteq2dv 5204	. . . . . . . . 9 ⊢ (𝑏 = 𝑗 → (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)))
17	16	eleq1d 2846	. . . . . . . 8 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝))
18		fveq1 6880	. . . . . . . . . 10 ⊢ (𝑐 = 𝑥 → (𝑐‘𝑗) = (𝑥‘𝑗))
19	18	cbvmptv 5214	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) = (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗))
20	19	eleq1i 2852	. . . . . . . 8 ⊢ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑗)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
21	17, 20	bitrdi 290	. . . . . . 7 ⊢ (𝑏 = 𝑗 → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝))
22	21	cbvralvw 3241	. . . . . 6 ⊢ (∀𝑏 ∈ 𝑉 (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) ∈ 𝑝 ↔ ∀𝑗 ∈ 𝑉 (𝑥 ∈ (ℤ ↑m 𝑉) ↦ (𝑥‘𝑗)) ∈ 𝑝)
23	14, 22	bitrdi 290	. . . . 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 643	. . . 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 642	. . 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 3432	. 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 43424	. 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 7443	. . . 4 ⊢ (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
29	28	pwex 5351	. . 3 ⊢ 𝒫 (ℤ ↑m (ℤ ↑m 𝑉)) ∈ V
30	29	rabex 5309	. 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 6989	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 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414  Vcvv 3453  𝒫 cpw 4561  {csn 4588   ↦ cmpt 5191   × cxp 5659  ‘cfv 6536  (class class class)co 7410   ∘_f cof 7672   ↑_m cmap 8823   + caddc 11102   · cmul 11104  ℤcz 12590  mzPolyCldcmzpcl 43422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-mzpcl 43424

This theorem is referenced by:  elmzpcl  43427