Theorem pm2mp 21436

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 2824	. . 3 ⊢ (0g‘𝐶) = (0g‘𝐶)
3		crngring 19311	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
4	3	anim2i 618	. . . . 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 21304	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
8		ringcmn 19334	. . . . 5 ⊢ (𝐶 ∈ Ring → 𝐶 ∈ CMnd)
9	4, 7, 8	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CMnd)
10	9	adantr 483	. . 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 21055	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
13	3, 12	sylan2 594	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ Ring)
14		monmat2matmon.q	. . . . . 6 ⊢ 𝑄 = (Poly1‘𝐴)
15	14	ply1ring 20419	. . . . 5 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
16		ringmnd 19309	. . . . 5 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
17	13, 15, 16	3syl 18	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑄 ∈ Mnd)
18	17	adantr 483	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑄 ∈ Mnd)
19		nn0ex 11906	. . . 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 2824	. . . . . . 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 21427	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
27	3, 26	sylan2 594	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
28	27	adantr 483	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝐼 ∈ (𝐶 GrpHom 𝑄))
29		ghmmhm 18371	. . . 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 483	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
32	31	adantr 483	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
33		elmapi 8431	. . . . . . 7 ⊢ (𝑀 ∈ (𝐾 ↑m ℕ0) → 𝑀:ℕ0⟶𝐾)
34	33	adantr 483	. . . . . 6 ⊢ ((𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴)) → 𝑀:ℕ0⟶𝐾)
35	34	adantl 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → 𝑀:ℕ0⟶𝐾)
36	35	ffvelrnda 6854	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → (𝑀‘𝑛) ∈ 𝐾)
37		simpr 487	. . . 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 21351	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ ((𝑀‘𝑛) ∈ 𝐾 ∧ 𝑛 ∈ ℕ0)) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
44	32, 36, 37, 43	syl12anc 834	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ 𝐵)
45		fvexd 6688	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) → (0g‘𝐶) ∈ V)
46		ovexd 7194	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ (𝐾 ↑m ℕ0) ∧ 𝑀 finSupp (0g‘𝐴))) ∧ 𝑛 ∈ ℕ0) → ((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) ∈ V)
47		simpr 487	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) → 𝑀 ∈ (𝐾 ↑m ℕ0))
48		fvex 6686	. . . . . . 7 ⊢ (0g‘𝐴) ∈ V
49		fsuppmapnn0ub 13366	. . . . . . 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 588	. . . . . 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 7203	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))))
52		csbov1g 7204	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌))
53		csbvarg 4386	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌𝑛 = 𝑥)
54	53	oveq1d 7174	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌𝑛𝐸𝑌) = (𝑥𝐸𝑌))
55	52, 54	eqtrd 2859	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) = (𝑥𝐸𝑌))
56		csbfv2g 6717	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)))
57		csbfv2g 6717	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘⦋𝑥 / 𝑛⦌𝑛))
58	53	fveq2d 6677	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ℕ0 → (𝑀‘⦋𝑥 / 𝑛⦌𝑛) = (𝑀‘𝑥))
59	57, 58	eqtrd 2859	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑀‘𝑛) = (𝑀‘𝑥))
60	59	fveq2d 6677	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ℕ0 → (𝑇‘⦋𝑥 / 𝑛⦌(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
61	56, 60	eqtrd 2859	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛)) = (𝑇‘(𝑀‘𝑥)))
62	55, 61	oveq12d 7177	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℕ0 → (⦋𝑥 / 𝑛⦌(𝑛𝐸𝑌) · ⦋𝑥 / 𝑛⦌(𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
63	51, 62	eqtrd 2859	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℕ0 → ⦋𝑥 / 𝑛⦌((𝑛𝐸𝑌) · (𝑇‘(𝑀‘𝑛))) = ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))))
64	63	adantl 484	. . . . . . . . . . . 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 483	. . . . . . . . . . 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 6673	. . . . . . . . . . . . 13 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → (𝑇‘(𝑀‘𝑥)) = (𝑇‘(0g‘𝐴)))
67	66	oveq2d 7175	. . . . . . . . . . . 12 ⊢ ((𝑀‘𝑥) = (0g‘𝐴) → ((𝑥𝐸𝑌) · (𝑇‘(𝑀‘𝑥))) = ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))))
68	39, 11, 38, 5, 6, 1	mat2pmatghm 21341	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
69	3, 68	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
70	69	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑇 ∈ (𝐴 GrpHom 𝐶))
71		ghmmhm 18371	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (𝐴 GrpHom 𝐶) → 𝑇 ∈ (𝐴 MndHom 𝐶))
72		eqid 2824	. . . . . . . . . . . . . . . 16 ⊢ (0g‘𝐴) = (0g‘𝐴)
73	72, 2	mhm0 17967	. . . . . . . . . . . . . . 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 7175	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = ((𝑥𝐸𝑌) · (0g‘𝐶)))
76	5	ply1ring 20419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
77	3, 76	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
78	6	matlmod 21041	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝐶 ∈ LMod)
79	77, 78	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ LMod)
80	79	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝐶 ∈ LMod)
81	77	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 ∈ Ring)
82		eqid 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
83	82	ringmgp 19306	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
84	81, 83	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (mulGrp‘𝑃) ∈ Mnd)
85	84	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
86		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
87	3	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
88		eqid 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑃) = (Base‘𝑃)
89	42, 5, 88	vr1cl 20388	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑅 ∈ Ring → 𝑌 ∈ (Base‘𝑃))
90	87, 89	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ (Base‘𝑃))
91	90	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → 𝑌 ∈ (Base‘𝑃))
92	82, 88	mgpbas 19248	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
93	92, 41	mulgnn0cl 18247	. . . . . . . . . . . . . . . 16 ⊢ (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑥 ∈ ℕ0 ∧ 𝑌 ∈ (Base‘𝑃)) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
94	85, 86, 91, 93	syl3anc 1367	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘𝑃))
95	5	ply1crng 20369	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ CRing)
96	6	matsca2 21032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
97	95, 96	sylan2 594	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝐶))
98	97	eqcomd 2830	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝐶) = 𝑃)
99	98	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Scalar‘𝐶) = 𝑃)
100	99	fveq2d 6677	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (Base‘(Scalar‘𝐶)) = (Base‘𝑃))
101	94, 100	eleqtrrd 2919	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶)))
102		eqid 2824	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
103		eqid 2824	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
104	102, 40, 103, 2	lmodvs0 19671	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ LMod ∧ (𝑥𝐸𝑌) ∈ (Base‘(Scalar‘𝐶))) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
105	80, 101, 104	syl2anc 586	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (0g‘𝐶)) = (0g‘𝐶))
106	75, 105	eqtrd 2859	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ (𝐾 ↑m ℕ0)) ∧ 𝑦 ∈ ℕ0) ∧ 𝑥 ∈ ℕ0) → ((𝑥𝐸𝑌) · (𝑇‘(0g‘𝐴))) = (0g‘𝐶))
107	67, 106	sylan9eqr 2881	. . . . . . . . . . 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 2859	. . . . . . . . . 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 415	. . . . . . . . 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 3180	. . . . . . 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 3277	. . . . . 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 457	. . . 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 13368	. . 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 19063	. 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 765	. . . . 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 21435	. . . . 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 834	. . . 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 5164	. . 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 7175	. 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 2861	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 398   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  Vcvv 3497  ⦋csb 3886   class class class wbr 5069   ↦ cmpt 5149  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ↑_m cmap 8409  Fincfn 8512   finSupp cfsupp 8836   < clt 10678  ℕ₀cn0 11900  Basecbs 16486  Scalarcsca 16571   ·_𝑠 cvsca 16572  0_gc0g 16716   Σ_g cgsu 16717  Mndcmnd 17914   MndHom cmhm 17957  ._gcmg 18227   GrpHom cghm 18358  CMndccmn 18909  mulGrpcmgp 19242  Ringcrg 19300  CRingccrg 19301  LModclmod 19637  var₁cv1 20347  Poly₁cpl1 20348   Mat cmat 21019   matToPolyMat cmat2pmat 21315   pMatToMatPoly cpm2mp 21403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-ot 4579  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-of 7412  df-ofr 7413  df-om 7584  df-1st 7692  df-2nd 7693  df-supp 7834  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-2o 8106  df-oadd 8109  df-er 8292  df-map 8411  df-pm 8412  df-ixp 8465  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-fsupp 8837  df-sup 8909  df-oi 8977  df-card 9371  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-nn 11642  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16488  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-plusg 16581  df-mulr 16582  df-sca 16584  df-vsca 16585  df-ip 16586  df-tset 16587  df-ple 16588  df-ds 16590  df-hom 16592  df-cco 16593  df-0g 16718  df-gsum 16719  df-prds 16724  df-pws 16726  df-mre 16860  df-mrc 16861  df-acs 16863  df-mgm 17855  df-sgrp 17904  df-mnd 17915  df-mhm 17959  df-submnd 17960  df-grp 18109  df-minusg 18110  df-sbg 18111  df-mulg 18228  df-subg 18279  df-ghm 18359  df-cntz 18450  df-cmn 18911  df-abl 18912  df-mgp 19243  df-ur 19255  df-ring 19302  df-cring 19303  df-subrg 19536  df-lmod 19639  df-lss 19707  df-sra 19947  df-rgmod 19948  df-assa 20088  df-ascl 20090  df-psr 20139  df-mvr 20140  df-mpl 20141  df-opsr 20143  df-psr1 20351  df-vr1 20352  df-ply1 20353  df-coe1 20354  df-dsmm 20879  df-frlm 20894  df-mamu 20998  df-mat 21020  df-mat2pmat 21318  df-decpmat 21374  df-pm2mp 21404

This theorem is referenced by:  cpmidpmat  21484  cpmadumatpoly  21494