Theorem pm2mp 21576

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 2738	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
3		crngring 19428	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
4	3	anim2i 620	. . . . 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 21443	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8		ringcmn 19453	. . . . 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 21194	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13	3, 12	sylan2 596	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
14		monmat2matmon.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝐴)
15	14	ply1ring 21023	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
16		ringmnd 19426	. . . . 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 11982	. . . 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 2738	. . . . . . 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 21567	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
27	3, 26	sylan2 596	. . . . 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 18486	. . . 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 8459	. . . . . . 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	ffvelrnda 6861	. . . 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 21491	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
44	32, 36, 37, 43	syl12anc 836	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
45		fvexd 6689	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (0g‘𝐶) ∈ V)
46		ovexd 7205	. . . 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 6687	. . . . . . 7 ⊢ (0g‘𝐴) ∈ V
49		fsuppmapnn0ub 13454	. . . . . . 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 589	. . . . . 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 7214	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))))
52		csbov1g 7215	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌))
53		csbvarg 4321	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌𝑛 = 𝑥)
54	53	oveq1d 7185	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌) = (𝑥𝐸𝑌))
55	52, 54	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (𝑥𝐸𝑌))
56		csbfv2g 6718	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)))
57		csbfv2g 6718	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘⦋𝑥 / 𝑛⦌𝑛))
58	53	fveq2d 6678	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑀‘⦋𝑥 / 𝑛⦌𝑛) = (𝑀‘𝑥))
59	57, 58	eqtrd 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘𝑥))
60	59	fveq2d 6678	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
61	56, 60	eqtrd 2773	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
62	55, 61	oveq12d 7188	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
63	51, 62	eqtrd 2773	. . . . . . . . . . . . 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 6674	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → (𝑇‘(𝑀‘𝑥)) = (𝑇‘(0g‘𝐴)))
67	66	oveq2d 7186	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))))
68	39, 11, 38, 5, 6, 1	mat2pmatghm 21481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
69	3, 68	sylan2 596	. . . . . . . . . . . . . . . 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 18486	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝐶) → 𝑇 ∈ (𝐴 MndHom 𝐶))
72		eqid 2738	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘𝐴)
73	72, 2	mhm0 18080	. . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = ((𝑥𝐸𝑌) · (0g‘𝐶)))
76	5	ply1ring 21023	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
77	3, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
78	6	matlmod 21180	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ LMod)
79	77, 78	sylan2 596	. . . . . . . . . . . . . . 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	77	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring)
82		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
83	82	ringmgp 19422	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
84	81, 83	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝑃) ∈ Mnd)
85	84	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
86		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
87	3	adantl 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
88		eqid 2738	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
89	42, 5, 88	vr1cl 20992	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
90	87, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ (Base‘𝑃))
91	90	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑌 ∈ (Base‘𝑃))
92	82, 88	mgpbas 19364	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
93	92, 41	mulgnn0cl 18362	. . . . . . . . . . . . . . . 16 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑥 ∈ ℕ0 ∧ 𝑌 ∈ (Base‘𝑃)) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
94	85, 86, 91, 93	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
95	5	ply1crng 20973	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
96	6	matsca2 21171	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
97	95, 96	sylan2 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
98	97	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝐶) = 𝑃)
99	98	ad3antrrr 730	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
100	99	fveq2d 6678	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
101	94, 100	eleqtrrd 2836	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶)))
102		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
103		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
104	102, 40, 103, 2	lmodvs0 19787	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶))) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
105	80, 101, 104	syl2anc 587	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
106	75, 105	eqtrd 2773	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = (0g‘𝐶))
107	67, 106	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) ∧ (𝑀‘𝑥) = (0g‘𝐴)) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = (0g‘𝐶))
108	65, 107	eqtrd 2773	. . . . . . . . . 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))
109	108	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)))
110	109	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))))
111	110	ralimdva 3091	. . . . . . 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))))
112	111	reximdva 3184	. . . . . 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))))
113	50, 112	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))))
114	113	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)))
115	45, 46, 114	mptnn0fsupp 13456	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑛 ∈ ℕ0 ↦ ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) finSupp (0g‘𝐶))
116	1, 2, 10, 18, 20, 30, 44, 115	gsummptmhm 19179	. 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)))))))
117		simpll 767	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
118	5, 6, 1, 21, 22, 23, 11, 38, 14, 25, 41, 42, 40, 39	monmat2matmon 21575	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) = ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))
119	117, 36, 37, 118	syl12anc 836	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝐼‘((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛)))) = ((𝑀‘𝑛) ∗ (𝑛 ↑ 𝑋)))
120	119	mpteq2dva 5125	. . 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))))
121	120	oveq2d 7186	. 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)))))
122	116, 121	eqtr3d 2775	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 1542   ∈ wcel 2114  ∀wral 3053  ∃wrex 3054  Vcvv 3398  ⦋csb 3790   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170   ↑_m cmap 8437  Fincfn 8555   finSupp cfsupp 8906   < clt 10753  ℕ₀cn0 11976  Basecbs 16586  Scalarcsca 16671   ·_𝑠 cvsca 16672  0_gc0g 16816   Σ_g cgsu 16817  Mndcmnd 18027   MndHom cmhm 18070  ._gcmg 18342   GrpHom cghm 18473  CMndccmn 19024  mulGrpcmgp 19358  Ringcrg 19416  CRingccrg 19417  LModclmod 19753  var₁cv1 20951  Poly₁cpl1 20952   Mat cmat 21158   matToPolyMat cmat2pmat 21455   pMatToMatPoly cpm2mp 21543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-ot 4525  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-ofr 7426  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-pm 8440  df-ixp 8508  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-sup 8979  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-uz 12325  df-fz 12982  df-fzo 13125  df-seq 13461  df-hash 13783  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-ip 16686  df-tset 16687  df-ple 16688  df-ds 16690  df-hom 16692  df-cco 16693  df-0g 16818  df-gsum 16819  df-prds 16824  df-pws 16826  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-submnd 18073  df-grp 18222  df-minusg 18223  df-sbg 18224  df-mulg 18343  df-subg 18394  df-ghm 18474  df-cntz 18565  df-cmn 19026  df-abl 19027  df-mgp 19359  df-ur 19371  df-ring 19418  df-cring 19419  df-subrg 19652  df-lmod 19755  df-lss 19823  df-sra 20063  df-rgmod 20064  df-dsmm 20548  df-frlm 20563  df-assa 20669  df-ascl 20671  df-psr 20722  df-mvr 20723  df-mpl 20724  df-opsr 20726  df-psr1 20955  df-vr1 20956  df-ply1 20957  df-coe1 20958  df-mamu 21137  df-mat 21159  df-mat2pmat 21458  df-decpmat 21514  df-pm2mp 21544

This theorem is referenced by:  cpmidpmat  21624  cpmadumatpoly  21634