Theorem mzpsubst 43194

Description: Substituting polynomials for the variables of a polynomial results in a polynomial. 𝐺 is expected to depend on 𝑦 and provide the polynomials which are being substituted. (Contributed by Stefan O'Rear, 5-Oct-2014.)

Assertion

Ref	Expression
mzpsubst	⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))

Distinct variable groups:   𝑥,𝑊,𝑦   𝑥,𝐹   𝑥,𝑉,𝑦   𝑥,𝐺

Allowed substitution hints:   𝐹(𝑦)   𝐺(𝑦)

Proof of Theorem mzpsubst

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝑊 ∈ V)
2		elfvex 6869	. . 3 ⊢ (𝐹 ∈ (mzPoly‘𝑉) → 𝑉 ∈ V)
3	2	3ad2ant2 1135	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝑉 ∈ V)
4		simp3 1139	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
5		simp2 1138	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝐹 ∈ (mzPoly‘𝑉))
6		simpr 484	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑥 ∈ (ℤ ↑m 𝑊))
7		simpll3 1216	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
8		simpll2 1215	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑉 ∈ V)
9		mzpf 43182	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (mzPoly‘𝑊) → 𝐺:(ℤ ↑m 𝑊)⟶ℤ)
10	9	ffvelcdmda 7030	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝐺‘𝑥) ∈ ℤ)
11	10	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (𝐺 ∈ (mzPoly‘𝑊) → (𝐺‘𝑥) ∈ ℤ))
12	11	ralimdv 3152	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ))
13	12	imp 406	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ)
14		eqid 2737	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) = (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))
15	14	fmpt 7056	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
16	13, 15	sylib 218	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
17	16	adantr 480	. . . . . . . 8 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
18		zex 12524	. . . . . . . . 9 ⊢ ℤ ∈ V
19		simpr 484	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → 𝑉 ∈ V)
20		elmapg 8779	. . . . . . . . 9 ⊢ ((ℤ ∈ V ∧ 𝑉 ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ))
21	18, 19, 20	sylancr 588	. . . . . . . 8 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ))
22	17, 21	mpbird 257	. . . . . . 7 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))
23	6, 7, 8, 22	syl21anc 838	. . . . . 6 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))
24		vex 3434	. . . . . . 7 ⊢ 𝑏 ∈ V
25	24	fvconst2 7152	. . . . . 6 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) → (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = 𝑏)
26	23, 25	syl 17	. . . . 5 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = 𝑏)
27	26	mpteq2dva 5179	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ 𝑏))
28		mzpconstmpt 43186	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ 𝑏) ∈ (mzPoly‘𝑊))
29	28	3ad2antl1 1187	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ 𝑏) ∈ (mzPoly‘𝑊))
30	27, 29	eqeltrd 2837	. . 3 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
31		simpr 484	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑥 ∈ (ℤ ↑m 𝑊))
32		simpll3 1216	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
33		simpll2 1215	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑉 ∈ V)
34	31, 32, 33, 22	syl21anc 838	. . . . . . . 8 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))
35		fveq1 6833	. . . . . . . . 9 ⊢ (𝑐 = (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) → (𝑐‘𝑏) = ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏))
36		eqid 2737	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))
37		fvex 6847	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) ∈ V
38	35, 36, 37	fvmpt 6941	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏))
39	34, 38	syl 17	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏))
40		simplr 769	. . . . . . . 8 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑏 ∈ 𝑉)
41		fvex 6847	. . . . . . . 8 ⊢ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V
42		csbeq1 3841	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑦⦌𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
43	42	fveq1d 6836	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (⦋𝑎 / 𝑦⦌𝐺‘𝑥) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
44		nfcv 2899	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝐺‘𝑥)
45		nfcsb1v 3862	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋𝑎 / 𝑦⦌𝐺
46		nfcv 2899	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑥
47	45, 46	nffv 6844	. . . . . . . . . 10 ⊢ Ⅎ𝑦(⦋𝑎 / 𝑦⦌𝐺‘𝑥)
48		csbeq1a 3852	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → 𝐺 = ⦋𝑎 / 𝑦⦌𝐺)
49	48	fveq1d 6836	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑥) = (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
50	44, 47, 49	cbvmpt 5188	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) = (𝑎 ∈ 𝑉 ↦ (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
51	43, 50	fvmptg 6939	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑉 ∧ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
52	40, 41, 51	sylancl 587	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
53	39, 52	eqtrd 2772	. . . . . 6 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
54	53	mpteq2dva 5179	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (⦋𝑏 / 𝑦⦌𝐺‘𝑥)))
55		simpr 484	. . . . . . . 8 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
56		simpl3 1195	. . . . . . . 8 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
57		nfcsb1v 3862	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺
58	57	nfel1 2916	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)
59		csbeq1a 3852	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → 𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
60	59	eleq1d 2822	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝐺 ∈ (mzPoly‘𝑊) ↔ ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)))
61	58, 60	rspc 3553	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑉 → (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) → ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)))
62	55, 56, 61	sylc 65	. . . . . . 7 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊))
63		mzpf 43182	. . . . . . 7 ⊢ (⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊) → ⦋𝑏 / 𝑦⦌𝐺:(ℤ ↑m 𝑊)⟶ℤ)
64	62, 63	syl 17	. . . . . 6 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ⦋𝑏 / 𝑦⦌𝐺:(ℤ ↑m 𝑊)⟶ℤ)
65	64	feqmptd 6902	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ⦋𝑏 / 𝑦⦌𝐺 = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (⦋𝑏 / 𝑦⦌𝐺‘𝑥)))
66	54, 65	eqtr4d 2775	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = ⦋𝑏 / 𝑦⦌𝐺)
67	66, 62	eqeltrd 2837	. . 3 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
68		simp2l 1201	. . . . . 6 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → 𝑏:(ℤ ↑m 𝑉)⟶ℤ)
69	68	ffnd 6663	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → 𝑏 Fn (ℤ ↑m 𝑉))
70		simp3l 1203	. . . . . 6 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → 𝑐:(ℤ ↑m 𝑉)⟶ℤ)
71	70	ffnd 6663	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → 𝑐 Fn (ℤ ↑m 𝑉))
72		simp13 1207	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
73		simp12 1206	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → 𝑉 ∈ V)
74		simplll 775	. . . . . . 7 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑏 Fn (ℤ ↑m 𝑉))
75		simpllr 776	. . . . . . 7 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑐 Fn (ℤ ↑m 𝑉))
76		ovexd 7395	. . . . . . 7 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (ℤ ↑m 𝑉) ∈ V)
77		simpr 484	. . . . . . . . . 10 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑥 ∈ (ℤ ↑m 𝑊))
78		simplrl 777	. . . . . . . . . 10 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
79	77, 78, 12	sylc 65	. . . . . . . . 9 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ)
80	79, 15	sylib 218	. . . . . . . 8 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
81		simplrr 778	. . . . . . . . 9 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑉 ∈ V)
82	18, 81, 20	sylancr 588	. . . . . . . 8 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ))
83	80, 82	mpbird 257	. . . . . . 7 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))
84		fnfvof 7641	. . . . . . 7 ⊢ (((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ ((ℤ ↑m 𝑉) ∈ V ∧ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))) → ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
85	74, 75, 76, 83, 84	syl22anc 839	. . . . . 6 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
86	85	mpteq2dva 5179	. . . . 5 ⊢ (((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))))
87	69, 71, 72, 73, 86	syl22anc 839	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))))
88		simp2r 1202	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
89		simp3r 1204	. . . . 5 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
90		mzpaddmpt 43187	. . . . 5 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))) ∈ (mzPoly‘𝑊))
91	88, 89, 90	syl2anc 585	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) + (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))) ∈ (mzPoly‘𝑊))
92	87, 91	eqeltrd 2837	. . 3 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
93		fnfvof 7641	. . . . . . 7 ⊢ (((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ ((ℤ ↑m 𝑉) ∈ V ∧ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉))) → ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
94	74, 75, 76, 83, 93	syl22anc 839	. . . . . 6 ⊢ ((((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
95	94	mpteq2dva 5179	. . . . 5 ⊢ (((𝑏 Fn (ℤ ↑m 𝑉) ∧ 𝑐 Fn (ℤ ↑m 𝑉)) ∧ (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑉 ∈ V)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))))
96	69, 71, 72, 73, 95	syl22anc 839	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))))
97		mzpmulmpt 43188	. . . . 5 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))) ∈ (mzPoly‘𝑊))
98	88, 89, 97	syl2anc 585	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) · (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))) ∈ (mzPoly‘𝑊))
99	96, 98	eqeltrd 2837	. . 3 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ (𝑏:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)) ∧ (𝑐:(ℤ ↑m 𝑉)⟶ℤ ∧ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
100		fveq1 6833	. . . . 5 ⊢ (𝑎 = ((ℤ ↑m 𝑉) × {𝑏}) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
101	100	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = ((ℤ ↑m 𝑉) × {𝑏}) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
102	101	eleq1d 2822	. . 3 ⊢ (𝑎 = ((ℤ ↑m 𝑉) × {𝑏}) → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
103		fveq1 6833	. . . . 5 ⊢ (𝑎 = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
104	103	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
105	104	eleq1d 2822	. . 3 ⊢ (𝑎 = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
106		fveq1 6833	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
107	106	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
108	107	eleq1d 2822	. . 3 ⊢ (𝑎 = 𝑏 → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
109		fveq1 6833	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
110	109	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
111	110	eleq1d 2822	. . 3 ⊢ (𝑎 = 𝑐 → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
112		fveq1 6833	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f + 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
113	112	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = (𝑏 ∘f + 𝑐) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
114	113	eleq1d 2822	. . 3 ⊢ (𝑎 = (𝑏 ∘f + 𝑐) → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
115		fveq1 6833	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f · 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
116	115	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = (𝑏 ∘f · 𝑐) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
117	116	eleq1d 2822	. . 3 ⊢ (𝑎 = (𝑏 ∘f · 𝑐) → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
118		fveq1 6833	. . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
119	118	mpteq2dv 5180	. . . 4 ⊢ (𝑎 = 𝐹 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
120	119	eleq1d 2822	. . 3 ⊢ (𝑎 = 𝐹 → ((𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊) ↔ (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊)))
121	30, 67, 92, 99, 102, 105, 108, 111, 114, 117, 120	mzpindd 43192	. 2 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝐹 ∈ (mzPoly‘𝑉)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))
122	1, 3, 4, 5, 121	syl31anc 1376	1 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430  ⦋csb 3838  {csn 4568   ↦ cmpt 5167   × cxp 5622   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∘_f cof 7622   ↑_m cmap 8766   + caddc 11032   · cmul 11034  ℤcz 12515  mzPolycmzp 43168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-map 8768  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-mzpcl 43169  df-mzp 43170

This theorem is referenced by:  mzprename  43195