Theorem pm2mp 23123

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 20452	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
4	3	anim2i 629	. . . . 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 22990	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8		ringcmn 20491	. . . . 5 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
9	4, 7, 8	3syl 19	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CMnd)
10	9	adantr 486	. . 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 22738	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13	3, 12	sylan2 605	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
14		monmat2matmon.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝐴)
15	14	ply1ring 22545	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
16		ringmnd 20450	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
17	13, 15, 16	3syl 19	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑄 ∈ Mnd)
18	17	adantr 486	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑄 ∈ Mnd)
19		nn0ex 12593	. . . 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 23114	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
27	3, 26	sylan2 605	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
28	27	adantr 486	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
29		ghmmhm 19420	. . . 4 ⊢ (𝐼 ∈ (𝐶 GrpHom 𝑄) → 𝐼 ∈ (𝐶 MndHom 𝑄))
30	28, 29	syl 18	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 MndHom 𝑄))
31	4	adantr 486	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31	adantr 486	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
33		elmapi 8853	. . . . . . 7 ⊢ (𝑀 ∈ (𝐾 ↑m ℕ0) → 𝑀:ℕ0⟶𝐾)
34	33	adantr 486	. . . . . 6 ⊢ ((𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴)) → 𝑀:ℕ0⟶𝐾)
35	34	adantl 487	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑀:ℕ0⟶𝐾)
36	35	ffvelcdmda 7076	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑀‘𝑛) ∈ 𝐾)
37		simpr 490	. . . 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 23038	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
44	32, 36, 37, 43	syl12anc 850	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
45		fvexd 6892	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (0g‘𝐶) ∈ V)
46		ovexd 7447	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ V)
47		simpr 490	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → 𝑀 ∈ (𝐾 ↑m ℕ0))
48		fvex 6890	. . . . . . 7 ⊢ (0g‘𝐴) ∈ V
49		fsuppmapnn0ub 14118	. . . . . . 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 598	. . . . . 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 7458	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))))
52		csbov1g 7459	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌))
53		csbvarg 4392	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌𝑛 = 𝑥)
54	53	oveq1d 7427	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌) = (𝑥𝐸𝑌))
55	52, 54	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (𝑥𝐸𝑌))
56		csbfv2g 6923	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)))
57		csbfv2g 6923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘⦋𝑥 / 𝑛⦌𝑛))
58	53	fveq2d 6881	. . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 486	. . . . . . . . . . 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 6877	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → (𝑇‘(𝑀‘𝑥)) = (𝑇‘(0g‘𝐴)))
67	66	oveq2d 7428	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))))
68	39, 11, 38, 5, 6, 1	mat2pmatghm 23028	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
69	3, 68	sylan2 605	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
70	69	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
71		ghmmhm 19420	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝐶) → 𝑇 ∈ (𝐴 MndHom 𝐶))
72		eqid 2761	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘𝐴)
73	72, 2	mhm0 18969	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 MndHom 𝐶) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
74	70, 71, 73	3syl 19	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑇‘(0g‘𝐴)) = (0g‘𝐶))
75	74	oveq2d 7428	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = ((𝑥𝐸𝑌) · (0g‘𝐶)))
76	5	ply1ring 22545	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
77	3, 76	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
78	6	matlmod 22724	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ LMod)
79	77, 78	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ LMod)
80	79	ad3antrrr 743	. . . . . . . . . . . . . 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 20345	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
84	77	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring)
85	81	ringmgp 20445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
86	84, 85	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝑃) ∈ Mnd)
87	86	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
88		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
89	3	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
90	42, 5, 82	vr1cl 22515	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
91	89, 90	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ (Base‘𝑃))
92	91	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑌 ∈ (Base‘𝑃))
93	83, 41, 87, 88, 92	mulgnn0cld 19285	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
94	5	ply1crng 22496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
95	6	matsca2 22715	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
96	94, 95	sylan2 605	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
97	96	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝐶) = 𝑃)
98	97	ad3antrrr 743	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
99	98	fveq2d 6881	. . . . . . . . . . . . . . 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 21151	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶))) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
104	80, 100, 103	syl2anc 596	. . . . . . . . . . . . 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 418	. . . . . . . . 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 58	. . . . . . . 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 3175	. . . . . . 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 3176	. . . . . 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 48	. . . . 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 460	. . . 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 14120	. . 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 20134	. 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 779	. . . . 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 23122	. . . . 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 850	. . . 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 5198	. . 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 7428	. 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)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451  ⦋csb 3847   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  Fincfn 8957   finSupp cfsupp 9337   < clt 11324  ℕ₀cn0 12587  Basecbs 17367  Scalarcsca 17411   ·_𝑠 cvsca 17412  0_gc0g 17590   Σ_g cgsu 17591  Mndcmnd 18903   MndHom cmhm 18956  ._gcmg 19257   GrpHom cghm 19407  CMndccmn 19974  mulGrpcmgp 20340  Ringcrg 20439  CRingccrg 20440  LModclmod 21115  var₁cv1 22474  Poly₁cpl1 22475   Mat cmat 22702   matToPolyMat cmat2pmat 23002   pMatToMatPoly cpm2mp 23090

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682  df-ofr 7683  df-om 7867  df-1st 7990  df-2nd 7991  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-pm 8834  df-ixp 8910  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-fsupp 9338  df-sup 9418  df-oi 9488  df-card 10001  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-nn 12317  df-2 12386  df-3 12387  df-4 12388  df-5 12389  df-6 12390  df-7 12391  df-8 12392  df-9 12393  df-n0 12588  df-z 12675  df-dec 12796  df-uz 12947  df-fz 13621  df-fzo 13769  df-seq 14125  df-hash 14455  df-struct 17305  df-sets 17322  df-slot 17340  df-ndx 17352  df-base 17368  df-ress 17389  df-plusg 17421  df-mulr 17422  df-sca 17424  df-vsca 17425  df-ip 17426  df-tset 17427  df-ple 17428  df-ds 17430  df-hom 17432  df-cco 17433  df-0g 17592  df-gsum 17593  df-prds 17598  df-pws 17600  df-mre 17736  df-mrc 17737  df-acs 17739  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-mhm 18958  df-submnd 18959  df-grp 19127  df-minusg 19128  df-sbg 19129  df-mulg 19258  df-subg 19313  df-ghm 19408  df-cntz 19511  df-cmn 19976  df-abl 19977  df-mgp 20341  df-rng 20355  df-ur 20388  df-ring 20441  df-cring 20442  df-subrng 20778  df-subrg 20802  df-lmod 21117  df-lss 21187  df-sra 21428  df-rgmod 21429  df-dsmm 22018  df-frlm 22033  df-assa 22141  df-ascl 22143  df-psr 22197  df-mvr 22198  df-mpl 22199  df-opsr 22201  df-psr1 22478  df-vr1 22479  df-ply1 22480  df-coe1 22481  df-mamu 22686  df-mat 22703  df-mat2pmat 23005  df-decpmat 23061  df-pm2mp 23091

This theorem is used by:  cpmidpmat  23171  cpmadumatpoly  23181