Theorem mzpsubst 43462

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 6918	. . 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 489	. . . . . . 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 43450	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (mzPoly‘𝑊) → 𝐺:(ℤ ↑m 𝑊)⟶ℤ)
10	9	ffvelcdmda 7081	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ (mzPoly‘𝑊) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → (𝐺‘𝑥) ∈ ℤ)
11	10	expcom 418	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (𝐺 ∈ (mzPoly‘𝑊) → (𝐺‘𝑥) ∈ ℤ))
12	11	ralimdv 3179	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ ↑m 𝑊) → (∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊) → ∀𝑦 ∈ 𝑉 (𝐺‘𝑥) ∈ ℤ))
13	12	imp 411	. . . . . . . . . 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 7107	. . . . . . . . . 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 485	. . . . . . . 8 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ)
18		zex 12601	. . . . . . . . 9 ⊢ ℤ ∈ V
19		simpr 489	. . . . . . . . 9 ⊢ (((𝑥 ∈ (ℤ ↑m 𝑊) ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑉 ∈ V) → 𝑉 ∈ V)
20		elmapg 8837	. . . . . . . . 9 ⊢ ((ℤ ∈ V ∧ 𝑉 ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) ∈ (ℤ ↑m 𝑉) ↔ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)):𝑉⟶ℤ))
21	18, 19, 20	sylancr 598	. . . . . . . 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 850	. . . . . 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 3459	. . . . . . 7 ⊢ 𝑏 ∈ V
25	24	fvconst2 7204	. . . . . 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 5205	. . . 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 43454	. . . . 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 2863	. . 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 489	. . . . . . . . 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 850	. . . . . . . 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 6882	. . . . . . . . 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 6896	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) ∈ V
38	35, 36, 37	fvmpt 6991	. . . . . . . 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 780	. . . . . . . 8 ⊢ ((((𝑊 ∈ V ∧ 𝑉 ∈ V ∧ ∀𝑦 ∈ 𝑉 𝐺 ∈ (mzPoly‘𝑊)) ∧ 𝑏 ∈ 𝑉) ∧ 𝑥 ∈ (ℤ ↑m 𝑊)) → 𝑏 ∈ 𝑉)
41		fvex 6896	. . . . . . . 8 ⊢ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V
42		csbeq1 3857	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ⦋𝑎 / 𝑦⦌𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
43	42	fveq1d 6885	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (⦋𝑎 / 𝑦⦌𝐺‘𝑥) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
44		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑎(𝐺‘𝑥)
45		nfcsb1v 3878	. . . . . . . . . . 11 ⊢ Ⅎ𝑦⦋𝑎 / 𝑦⦌𝐺
46		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑦𝑥
47	45, 46	nffv 6893	. . . . . . . . . 10 ⊢ Ⅎ𝑦(⦋𝑎 / 𝑦⦌𝐺‘𝑥)
48		csbeq1a 3868	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → 𝐺 = ⦋𝑎 / 𝑦⦌𝐺)
49	48	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝐺‘𝑥) = (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
50	44, 47, 49	cbvmpt 5214	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)) = (𝑎 ∈ 𝑉 ↦ (⦋𝑎 / 𝑦⦌𝐺‘𝑥))
51	43, 50	fvmptg 6989	. . . . . . . 8 ⊢ ((𝑏 ∈ 𝑉 ∧ (⦋𝑏 / 𝑦⦌𝐺‘𝑥) ∈ V) → ((𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))‘𝑏) = (⦋𝑏 / 𝑦⦌𝐺‘𝑥))
52	40, 41, 51	sylancl 597	. . . . . . 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 5205	. . . . 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 489	. . . . . . . 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 3878	. . . . . . . . . 10 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺
58	57	nfel1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)
59		csbeq1a 3868	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → 𝐺 = ⦋𝑏 / 𝑦⦌𝐺)
60	59	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝐺 ∈ (mzPoly‘𝑊) ↔ ⦋𝑏 / 𝑦⦌𝐺 ∈ (mzPoly‘𝑊)))
61	58, 60	rspc 3570	. . . . . . . 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 43450	. . . . . . 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 6951	. . . . 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 2863	. . 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 6708	. . . . 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 6708	. . . . 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 786	. . . . . . 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 787	. . . . . . 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 7447	. . . . . . 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 489	. . . . . . . . . 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 788	. . . . . . . . . 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 789	. . . . . . . . 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 598	. . . . . . . 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 7693	. . . . . . 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 851	. . . . . 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 5205	. . . . 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 851	. . . 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 43455	. . . . 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 595	. . . 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 2863	. . 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 7693	. . . . . . 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 851	. . . . . 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 5205	. . . . 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 851	. . . 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 43456	. . . . 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 595	. . . 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 2863	. . 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 6882	. . . . 5 ⊢ (𝑎 = ((ℤ ↑m 𝑉) × {𝑏}) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (((ℤ ↑m 𝑉) × {𝑏})‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
101	100	mpteq2dv 5206	. . . 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 2848	. . 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 6882	. . . . 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 5206	. . . 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 2848	. . 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 6882	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
107	106	mpteq2dv 5206	. . . 4 ⊢ (𝑎 = 𝑏 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑏‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
108	107	eleq1d 2848	. . 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 6882	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
110	109	mpteq2dv 5206	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑐‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
111	110	eleq1d 2848	. . 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 6882	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f + 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f + 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
113	112	mpteq2dv 5206	. . . 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 2848	. . 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 6882	. . . . 5 ⊢ (𝑎 = (𝑏 ∘f · 𝑐) → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = ((𝑏 ∘f · 𝑐)‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
116	115	mpteq2dv 5206	. . . 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 2848	. . 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 6882	. . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))) = (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥))))
119	118	mpteq2dv 5206	. . . 4 ⊢ (𝑎 = 𝐹 → (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝑎‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))) = (𝑥 ∈ (ℤ ↑m 𝑊) ↦ (𝐹‘(𝑦 ∈ 𝑉 ↦ (𝐺‘𝑥)))))
120	119	eleq1d 2848	. . 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 43460	. 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 1400	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  ⦋csb 3854  {csn 4590   ↦ cmpt 5193   × cxp 5661   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∘_f cof 7674   ↑_m cmap 8825   + caddc 11104   · cmul 11106  ℤcz 12592  mzPolycmzp 43436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7676  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506  df-z 12593  df-mzpcl 43437  df-mzp 43438

This theorem is referenced by:  mzprename  43463