Theorem pm2mp 22865

Description: The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-Nov-2019.) (Revised by AV, 7-Dec-2019.)

Hypotheses

Ref	Expression
monmat2matmon.p	⊢ 𝑃 = (Poly1‘𝑅)
monmat2matmon.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
monmat2matmon.b	⊢ 𝐵 = (Base‘𝐶)
monmat2matmon.m1	⊢ ∗ = ( ·𝑠 ‘𝑄)
monmat2matmon.e1	⊢ ↑ = (.g‘(mulGrp‘𝑄))
monmat2matmon.x	⊢ 𝑋 = (var1‘𝐴)
monmat2matmon.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
monmat2matmon.k	⊢ 𝐾 = (Base‘𝐴)
monmat2matmon.q	⊢ 𝑄 = (Poly1‘𝐴)
monmat2matmon.i	⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)
monmat2matmon.e2	⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
monmat2matmon.y	⊢ 𝑌 = (var1‘𝑅)
monmat2matmon.m2	⊢ · = ( ·𝑠 ‘𝐶)
monmat2matmon.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

Assertion

Ref	Expression
pm2mp	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))))

Distinct variable groups:   𝐴,𝑛   𝐵,𝑛   𝑛,𝐸   𝑛,𝐼   𝑛,𝐾   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑇,𝑛   𝑛,𝑌   · ,𝑛

Allowed substitution hints:   𝐶(𝑛)   𝑃(𝑛)   𝑄(𝑛)   ↑ (𝑛)   ∗ (𝑛)   𝑋(𝑛)

Proof of Theorem pm2mp

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		monmat2matmon.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2761	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
3		crngring 20274	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
4	3	anim2i 626	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
5		monmat2matmon.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
6		monmat2matmon.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
7	5, 6	pmatring 22732	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8		ringcmn 20311	. . . . 5 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
9	4, 7, 8	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CMnd)
10	9	adantr 484	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐶 ∈ CMnd)
11		monmat2matmon.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
12	11	matring 22483	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13	3, 12	sylan2 602	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
14		monmat2matmon.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝐴)
15	14	ply1ring 22289	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
16		ringmnd 20272	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
17	13, 15, 16	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑄 ∈ Mnd)
18	17	adantr 484	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑄 ∈ Mnd)
19		nn0ex 12484	. . . 4 ⊢ ℕ0 ∈ V
20	19	a1i 11	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → ℕ0 ∈ V)
21		monmat2matmon.m1	. . . . . . 7 ⊢ ∗ = ( ·𝑠 ‘𝑄)
22		monmat2matmon.e1	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
23		monmat2matmon.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝐴)
24		eqid 2761	. . . . . . 7 ⊢ (Base‘𝑄) = (Base‘𝑄)
25		monmat2matmon.i	. . . . . . 7 ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)
26	5, 6, 1, 21, 22, 23, 11, 14, 24, 25	pm2mpghm 22856	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
27	3, 26	sylan2 602	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
28	27	adantr 484	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
29		ghmmhm 19249	. . . 4 ⊢ (𝐼 ∈ (𝐶 GrpHom 𝑄) → 𝐼 ∈ (𝐶 MndHom 𝑄))
30	28, 29	syl 17	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 MndHom 𝑄))
31	4	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31	adantr 484	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
33		elmapi 8826	. . . . . . 7 ⊢ (𝑀 ∈ (𝐾 ↑m ℕ0) → 𝑀:ℕ0⟶𝐾)
34	33	adantr 484	. . . . . 6 ⊢ ((𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴)) → 𝑀:ℕ0⟶𝐾)
35	34	adantl 485	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑀:ℕ0⟶𝐾)
36	35	ffvelcdmda 7061	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑀‘𝑛) ∈ 𝐾)
37		simpr 488	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
38		monmat2matmon.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐴)
39		monmat2matmon.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
40		monmat2matmon.m2	. . . . 5 ⊢ · = ( ·𝑠 ‘𝐶)
41		monmat2matmon.e2	. . . . 5 ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
42		monmat2matmon.y	. . . . 5 ⊢ 𝑌 = (var1‘𝑅)
43	11, 38, 39, 5, 6, 1, 40, 41, 42	mat2pmatscmxcl 22780	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
44	32, 36, 37, 43	syl12anc 847	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
45		fvexd 6878	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (0g‘𝐶) ∈ V)
46		ovexd 7427	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ V)
47		simpr 488	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → 𝑀 ∈ (𝐾 ↑m ℕ0))
48		fvex 6876	. . . . . . 7 ⊢ (0g‘𝐴) ∈ V
49		fsuppmapnn0ub 14005	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐾 ↑m ℕ0) ∧ (0g‘𝐴) ∈ V) → (𝑀 finSupp (0g‘𝐴) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑀‘𝑥) = (0g‘𝐴))))
50	47, 48, 49	sylancl 595	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → (𝑀 finSupp (0g‘𝐴) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑀‘𝑥) = (0g‘𝐴))))
51		csbov12g 7438	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))))
52		csbov1g 7439	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌))
53		csbvarg 4387	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌𝑛 = 𝑥)
54	53	oveq1d 7407	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌) = (𝑥𝐸𝑌))
55	52, 54	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (𝑥𝐸𝑌))
56		csbfv2g 6909	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)))
57		csbfv2g 6909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘⦋𝑥 / 𝑛⦌𝑛))
58	53	fveq2d 6867	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑀‘⦋𝑥 / 𝑛⦌𝑛) = (𝑀‘𝑥))
59	57, 58	eqtrd 2796	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘𝑥))
60	59	fveq2d 6867	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
61	56, 60	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
62	55, 61	oveq12d 7410	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
63	51, 62	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
64	63	adantl 485	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
65	64	adantr 484	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑀‘𝑥) = (0g‘𝐴)) → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
66		fveq2 6863	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → (𝑇‘(𝑀‘𝑥)) = (𝑇‘(0g‘𝐴)))
67	66	oveq2d 7408	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))))
68	39, 11, 38, 5, 6, 1	mat2pmatghm 22770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
69	3, 68	sylan2 602	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
70	69	ad3antrrr 740	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
71		ghmmhm 19249	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝐶) → 𝑇 ∈ (𝐴 MndHom 𝐶))
72		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘𝐴)
73	72, 2	mhm0 18811	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 MndHom 𝐶) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
74	70, 71, 73	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
75	74	oveq2d 7408	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = ((𝑥𝐸𝑌) · (0g‘𝐶)))
76	5	ply1ring 22289	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
77	3, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
78	6	matlmod 22469	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ LMod)
79	77, 78	sylan2 602	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ LMod)
80	79	ad3antrrr 740	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐶 ∈ LMod)
81		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
82		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑃) = (Base‘𝑃)
83	81, 82	mgpbas 20174	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
84	77	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring)
85	81	ringmgp 20268	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝑃) ∈ Mnd)
87	86	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
88		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
89	3	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
90	42, 5, 82	vr1cl 22259	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ (Base‘𝑃))
92	91	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑌 ∈ (Base‘𝑃))
93	83, 41, 87, 88, 92	mulgnn0cld 19120	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
94	5	ply1crng 22240	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
95	6	matsca2 22460	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
96	94, 95	sylan2 602	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
97	96	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝐶) = 𝑃)
98	97	ad3antrrr 740	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
99	98	fveq2d 6867	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
100	93, 99	eleqtrrd 2864	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶)))
101		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
102		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
103	101, 40, 102, 2	lmodvs0 20943	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶))) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
104	80, 100, 103	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
105	75, 104	eqtrd 2796	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = (0g‘𝐶))
106	67, 105	sylan9eqr 2818	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑀‘𝑥) = (0g‘𝐴)) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = (0g‘𝐶))
107	65, 106	eqtrd 2796	. . . . . . . . . 10 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑀‘𝑥) = (0g‘𝐴)) → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶))
108	107	ex 416	. . . . . . . . 9 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑀‘𝑥) = (0g‘𝐴) → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶)))
109	108	imim2d 57	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → (𝑀‘𝑥) = (0g‘𝐴)) → (𝑦 < 𝑥 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶))))
110	109	ralimdva 3173	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑀‘𝑥) = (0g‘𝐴)) → ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶))))
111	110	reximdva 3174	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → (∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑀‘𝑥) = (0g‘𝐴)) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶))))
112	50, 111	syld 47	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → (𝑀 finSupp (0g‘𝐴) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶))))
113	112	impr 458	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → ∃𝑦 ∈ ℕ0 ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (0g‘𝐶)))
114	45, 46, 113	mptnn0fsupp 14007	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) finSupp (0g‘𝐶))
115	1, 2, 10, 18, 20, 30, 44, 114	gsummptmhm 19963	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))))
116		simpll 776	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
117	5, 6, 1, 21, 22, 23, 11, 38, 14, 25, 41, 42, 40, 39	monmat2matmon 22864	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) = ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))
118	116, 36, 37, 117	syl12anc 847	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) = ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))
119	118	mpteq2dva 5192	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑛 ∈ ℕ0 ↦ (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))))) = (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋))))
120	119	oveq2d 7408	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑄 Σg (𝑛 ∈ ℕ0 ↦ (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))))
121	115, 120	eqtr3d 2798	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝐼‘(𝐶 Σg (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))))) = (𝑄 Σg (𝑛 ∈ ℕ0 ↦ ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  Vcvv 3453  ⦋csb 3852   class class class wbr 5099   ↦ cmpt 5180  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803  Fincfn 8923   finSupp cfsupp 9304   < clt 11213  ℕ₀cn0 12478  Basecbs 17228  Scalarcsca 17272   ·_𝑠 cvsca 17273  0_gc0g 17451   Σ_g cgsu 17452  Mndcmnd 18751   MndHom cmhm 18798  ._gcmg 19092   GrpHom cghm 19236  CMndccmn 19803  mulGrpcmgp 20169  Ringcrg 20262  CRingccrg 20263  LModclmod 20907  var₁cv1 22218  Poly₁cpl1 22219   Mat cmat 22447   matToPolyMat cmat2pmat 22744   pMatToMatPoly cpm2mp 22832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-iin 4951  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-of 7656  df-ofr 7657  df-om 7843  df-1st 7966  df-2nd 7967  df-supp 8136  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-er 8673  df-map 8805  df-pm 8806  df-ixp 8876  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-fsupp 9305  df-sup 9385  df-oi 9455  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-uz 12837  df-fz 13510  df-fzo 13657  df-seq 14012  df-hash 14341  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-plusg 17282  df-mulr 17283  df-sca 17285  df-vsca 17286  df-ip 17287  df-tset 17288  df-ple 17289  df-ds 17291  df-hom 17293  df-cco 17294  df-0g 17453  df-gsum 17454  df-prds 17459  df-pws 17461  df-mre 17597  df-mrc 17598  df-acs 17600  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-mhm 18800  df-submnd 18801  df-grp 18961  df-minusg 18962  df-sbg 18963  df-mulg 19093  df-subg 19148  df-ghm 19237  df-cntz 19340  df-cmn 19805  df-abl 19806  df-mgp 20170  df-rng 20182  df-ur 20211  df-ring 20264  df-cring 20265  df-subrng 20575  df-subrg 20599  df-lmod 20909  df-lss 20979  df-sra 21220  df-rgmod 21221  df-dsmm 21764  df-frlm 21779  df-assa 21885  df-ascl 21887  df-psr 21941  df-mvr 21942  df-mpl 21943  df-opsr 21945  df-psr1 22222  df-vr1 22223  df-ply1 22224  df-coe1 22225  df-mamu 22431  df-mat 22448  df-mat2pmat 22747  df-decpmat 22803  df-pm2mp 22833

This theorem is referenced by:  cpmidpmat  22913  cpmadumatpoly  22923