Theorem pm2mp 22847

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 2735	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
3		crngring 20263	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
4	3	anim2i 617	. . . . 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 22714	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8		ringcmn 20296	. . . . 5 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
9	4, 7, 8	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CMnd)
10	9	adantr 480	. . 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 22465	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13	3, 12	sylan2 593	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
14		monmat2matmon.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝐴)
15	14	ply1ring 22265	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
16		ringmnd 20261	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
17	13, 15, 16	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑄 ∈ Mnd)
18	17	adantr 480	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑄 ∈ Mnd)
19		nn0ex 12530	. . . 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 2735	. . . . . . 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 22838	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
27	3, 26	sylan2 593	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
28	27	adantr 480	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
29		ghmmhm 19257	. . . 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 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31	adantr 480	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
33		elmapi 8888	. . . . . . 7 ⊢ (𝑀 ∈ (𝐾 ↑m ℕ0) → 𝑀:ℕ0⟶𝐾)
34	33	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴)) → 𝑀:ℕ0⟶𝐾)
35	34	adantl 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑀:ℕ0⟶𝐾)
36	35	ffvelcdmda 7104	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑀‘𝑛) ∈ 𝐾)
37		simpr 484	. . . 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 22762	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
44	32, 36, 37, 43	syl12anc 837	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
45		fvexd 6922	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (0g‘𝐶) ∈ V)
46		ovexd 7466	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ V)
47		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → 𝑀 ∈ (𝐾 ↑m ℕ0))
48		fvex 6920	. . . . . . 7 ⊢ (0g‘𝐴) ∈ V
49		fsuppmapnn0ub 14033	. . . . . . 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 586	. . . . . 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 7477	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))))
52		csbov1g 7478	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌))
53		csbvarg 4440	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌𝑛 = 𝑥)
54	53	oveq1d 7446	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌) = (𝑥𝐸𝑌))
55	52, 54	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (𝑥𝐸𝑌))
56		csbfv2g 6956	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)))
57		csbfv2g 6956	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘⦋𝑥 / 𝑛⦌𝑛))
58	53	fveq2d 6911	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑀‘⦋𝑥 / 𝑛⦌𝑛) = (𝑀‘𝑥))
59	57, 58	eqtrd 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘𝑥))
60	59	fveq2d 6911	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
61	56, 60	eqtrd 2775	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
62	55, 61	oveq12d 7449	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
63	51, 62	eqtrd 2775	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
64	63	adantl 481	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 6907	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → (𝑇‘(𝑀‘𝑥)) = (𝑇‘(0g‘𝐴)))
67	66	oveq2d 7447	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))))
68	39, 11, 38, 5, 6, 1	mat2pmatghm 22752	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
69	3, 68	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
70	69	ad3antrrr 730	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
71		ghmmhm 19257	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝐶) → 𝑇 ∈ (𝐴 MndHom 𝐶))
72		eqid 2735	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘𝐴)
73	72, 2	mhm0 18820	. . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = ((𝑥𝐸𝑌) · (0g‘𝐶)))
76	5	ply1ring 22265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
77	3, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
78	6	matlmod 22451	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ LMod)
79	77, 78	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ LMod)
80	79	ad3antrrr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐶 ∈ LMod)
81		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
82		eqid 2735	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑃) = (Base‘𝑃)
83	81, 82	mgpbas 20158	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
84	77	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring)
85	81	ringmgp 20257	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
86	84, 85	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝑃) ∈ Mnd)
87	86	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
88		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
89	3	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
90	42, 5, 82	vr1cl 22235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
91	89, 90	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ (Base‘𝑃))
92	91	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑌 ∈ (Base‘𝑃))
93	83, 41, 87, 88, 92	mulgnn0cld 19126	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
94	5	ply1crng 22216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
95	6	matsca2 22442	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
96	94, 95	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
97	96	eqcomd 2741	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝐶) = 𝑃)
98	97	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
99	98	fveq2d 6911	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
100	93, 99	eleqtrrd 2842	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶)))
101		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
102		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
103	101, 40, 102, 2	lmodvs0 20911	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶))) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
104	80, 100, 103	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
105	75, 104	eqtrd 2775	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = (0g‘𝐶))
106	67, 105	sylan9eqr 2797	. . . . . . . . . . 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 2775	. . . . . . . . . 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 412	. . . . . . . . 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 3165	. . . . . . 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 3166	. . . . . 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 454	. . . 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 14035	. . 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 19973	. 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 767	. . . . 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 22846	. . . . 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 837	. . . 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 5248	. . 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 7447	. 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 2777	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 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  Vcvv 3478  ⦋csb 3908   class class class wbr 5148   ↦ cmpt 5231  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865  Fincfn 8984   finSupp cfsupp 9399   < clt 11293  ℕ₀cn0 12524  Basecbs 17245  Scalarcsca 17301   ·_𝑠 cvsca 17302  0_gc0g 17486   Σ_g cgsu 17487  Mndcmnd 18760   MndHom cmhm 18807  ._gcmg 19098   GrpHom cghm 19243  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252  LModclmod 20875  var₁cv1 22193  Poly₁cpl1 22194   Mat cmat 22427   matToPolyMat cmat2pmat 22726   pMatToMatPoly cpm2mp 22814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-assa 21891  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-mamu 22411  df-mat 22428  df-mat2pmat 22729  df-decpmat 22785  df-pm2mp 22815

This theorem is referenced by:  cpmidpmat  22895  cpmadumatpoly  22905