Theorem mzpsubst 44019

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 1154	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝑊 ∈ V)
2		elfvex 6949	. . 3 ⊢ (𝐹 ∈ (mzPoly‘𝑉) → 𝑉 ∈ V)
3	2	3ad2ant2 1152	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝑉 ∈ V)
4		simp3 1156	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
5		simp2 1155	. 2 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → 𝐹 ∈ (mzPoly‘𝑉))
6		simpr 490	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑥 ∈ (ℤ ↑m 𝑊))
7		simpll3 1233	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
8		simpll2 1232	. . . . . . 7 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑉 ∈ V)
9		mzpf 44007	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (mzPoly‘𝑊) → 𝐺:(ℤ ↑m 𝑊)⟶ℤ)
10	9	ffvelcdmda 7114	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝐺‘𝑥) ∈ ℤ)
11	10	expcom 419	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (𝐺 ∈ (mzPoly‘𝑊) → (𝐺‘𝑥) ∈ ℤ))
12	11	ralimdv 3180	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ))
13	12	imp 412	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ)
14		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) = (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))
15	14	fmpt 7140	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
16	13, 15	sylib 221	. . . . . . . . 9 ⊢ ((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
17	16	adantr 486	. . . . . . . 8 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
18		zex 12740	. . . . . . . . 9 ⊢ ℤ ∈ V
19		simpr 490	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → 𝑉 ∈ V)
20		elmapg 8895	. . . . . . . . 9 ⊢ ((ℤ ∈ V ∧ 𝑉 ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ))
21	18, 19, 20	sylancr 599	. . . . . . . 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 260	. . . . . . 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 851	. . . . . 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 3458	. . . . . . 7 ⊢ 𝑏 ∈ V
25	24	fvconst2 7239	. . . . . 6 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) → (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = 𝑏)
26	23, 25	syl 18	. . . . 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 5202	. . . 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 44011	. . . . 5 ⊢ ((𝑊 ∈ V ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ 𝑏) ∈ (mzPoly‘𝑊))
29	28	3ad2antl1 1204	. . . 4 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ ℤ) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ 𝑏) ∈ (mzPoly‘𝑊))
30	27, 29	eqeltrd 2864	. . 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 490	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑥 ∈ (ℤ ↑m 𝑊))
32		simpll3 1233	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
33		simpll2 1232	. . . . . . . . 9 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑉 ∈ V)
34	31, 32, 33, 22	syl21anc 851	. . . . . . . 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 6913	. . . . . . . . 9 ⊢ (𝑐 = (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) → (𝑐‘𝑏) = ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏))
36		eqid 2763	. . . . . . . . 9 ⊢ (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏)) = (𝑐 ∈ (ℤ ↑m 𝑉) ↦ (𝑐‘𝑏))
37		fvex 6927	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) ∈ V
38	35, 36, 37	fvmpt 7022	. . . . . . . 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 18	. . . . . . 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 781	. . . . . . . 8 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑏 ∈ 𝑉)
41		fvex 6927	. . . . . . . 8 ⊢ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V
42		csbeq1 3854	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑦⦌𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
43	42	fveq1d 6916	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (⦋𝑎 / 𝑦⦌𝐺‘𝑥) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
44		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝐺‘𝑥)
45		nfcsb1v 3875	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋𝑎 / 𝑦⦌𝐺
46		nfcv 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑥
47	45, 46	nffv 6924	. . . . . . . . . 10 ⊢ Ⅎ𝑦(⦋𝑎 / 𝑦⦌𝐺‘𝑥)
48		csbeq1a 3865	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → 𝐺 = ⦋𝑎 / 𝑦⦌𝐺)
49	48	fveq1d 6916	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑥) = (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
50	44, 47, 49	cbvmpt 5211	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) = (𝑎 ∈ 𝑉 ↦ (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
51	43, 50	fvmptg 7020	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑉 ∧ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
52	40, 41, 51	sylancl 598	. . . . . . 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 2798	. . . . . 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 5202	. . . . 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 490	. . . . . . . 8 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
56		simpl3 1212	. . . . . . . 8 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊))
57		nfcsb1v 3875	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺
58	57	nfel1 2942	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)
59		csbeq1a 3865	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → 𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
60	59	eleq1d 2849	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝐺 ∈ (mzPoly‘𝑊) ↔ ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)))
61	58, 60	rspc 3568	. . . . . . . 8 ⊢ (𝑏 ∈ 𝑉 → (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) → ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)))
62	55, 56, 61	sylc 66	. . . . . . 7 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊))
63		mzpf 44007	. . . . . . 7 ⊢ (⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊) → ⦋𝑏 / 𝑦⦌𝐺:(ℤ ↑m 𝑊)⟶ℤ)
64	62, 63	syl 18	. . . . . 6 ⊢ (((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) → ⦋𝑏 / 𝑦⦌𝐺:(ℤ ↑m 𝑊)⟶ℤ)
65	64	feqmptd 6982	. . . . 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 2801	. . . 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 2864	. . 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 1218	. . . . . 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 6738	. . . . 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 1220	. . . . . 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 6738	. . . . 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 1224	. . . . 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 1223	. . . . 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 787	. . . . . . 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 788	. . . . . . 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 7484	. . . . . . 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 490	. . . . . . . . . 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 789	. . . . . . . . . 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 66	. . . . . . . . 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 221	. . . . . . . 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 790	. . . . . . . . 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 599	. . . . . . . 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 260	. . . . . . 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 7740	. . . . . . 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 852	. . . . . 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 5202	. . . . 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 852	. . . 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 1219	. . . . 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 1221	. . . . 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 44012	. . . . 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 596	. . . 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 2864	. . 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 7740	. . . . . . 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 852	. . . . . 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 5202	. . . . 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 852	. . . 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 44013	. . . . 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 596	. . . 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 2864	. . 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 6913	. . . . 5 ⊢ (𝑎 = ((ℤ ↑m 𝑉) × {𝑏}) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
101	100	mpteq2dv 5203	. . . 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 2849	. . 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 6913	. . . . 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 5203	. . . 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 2849	. . 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 6913	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
107	106	mpteq2dv 5203	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
108	107	eleq1d 2849	. . 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 6913	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
110	109	mpteq2dv 5203	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
111	110	eleq1d 2849	. . 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 6913	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f + 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
113	112	mpteq2dv 5203	. . . 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 2849	. . 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 6913	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f · 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
116	115	mpteq2dv 5203	. . . 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 2849	. . 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 6913	. . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
119	118	mpteq2dv 5203	. . . 4 ⊢ (𝑎 = 𝐹 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
120	119	eleq1d 2849	. . 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 44017	. 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 1401	1 ⊢ ((𝑊 ∈ V ∧ 𝐹 ∈ (mzPoly‘𝑉) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) ∈ (mzPoly‘𝑊))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  ⦋csb 3851  {csn 4588   ↦ cmpt 5190   × cxp 5654   Fn wfn 6559  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449   ∘_f cof 7721   ↑_m cmap 8883   + caddc 11240   · cmul 11242  ℤcz 12731  mzPolycmzp 43993

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-of 7723  df-om 7910  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-er 8753  df-map 8885  df-en 9010  df-dom 9011  df-sdom 9012  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-n0 12645  df-z 12732  df-mzpcl 43994  df-mzp 43995

This theorem is used by:  mzprename  44020