Theorem monmat2matmon 22718

Description: The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-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
monmat2matmon	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋)))

Proof of Theorem monmat2matmon

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		crngring 20161	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		simpll 766	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑁 ∈ Fin)
3		simplr 768	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑅 ∈ Ring)
4		monmat2matmon.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
5		monmat2matmon.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐴)
6		monmat2matmon.t	. . . . 5 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
7		monmat2matmon.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
8		monmat2matmon.c	. . . . 5 ⊢ 𝐶 = (𝑁 Mat 𝑃)
9		monmat2matmon.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
10		monmat2matmon.m2	. . . . 5 ⊢ · = ( ·𝑠 ‘𝐶)
11		monmat2matmon.e2	. . . . 5 ⊢ 𝐸 = (.g‘(mulGrp‘𝑃))
12		monmat2matmon.y	. . . . 5 ⊢ 𝑌 = (var1‘𝑅)
13	4, 5, 6, 7, 8, 9, 10, 11, 12	mat2pmatscmxcl 22634	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝐿𝐸𝑌) · (𝑇‘𝑀)) ∈ 𝐵)
14		monmat2matmon.m1	. . . . 5 ⊢ ∗ = ( ·𝑠 ‘𝑄)
15		monmat2matmon.e1	. . . . 5 ⊢ ↑ = (.g‘(mulGrp‘𝑄))
16		monmat2matmon.x	. . . . 5 ⊢ 𝑋 = (var1‘𝐴)
17		monmat2matmon.q	. . . . 5 ⊢ 𝑄 = (Poly1‘𝐴)
18		monmat2matmon.i	. . . . 5 ⊢ 𝐼 = (𝑁 pMatToMatPoly 𝑅)
19	7, 8, 9, 14, 15, 16, 4, 17, 18	pm2mpfval 22690	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ ((𝐿𝐸𝑌) · (𝑇‘𝑀)) ∈ 𝐵) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
20	2, 3, 13, 19	syl3anc 1373	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
21	1, 20	sylanl2 681	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))))
22		simpll 766	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
23		simpr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0))
24	23	anim1i 615	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0))
25		df-3an 1088	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ↔ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0))
26	24, 25	sylibr 234	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0))
27		eqid 2730	. . . . . . . . 9 ⊢ (0g‘𝐴) = (0g‘𝐴)
28	7, 8, 4, 5, 27, 11, 12, 10, 6	monmatcollpw 22673	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0)) → (((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) = if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)))
29	22, 26, 28	syl2anc 584	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) = if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)))
30	29	oveq1d 7405	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = (if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)))
31	1	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ0 → (𝑅 ∈ CRing → 𝑅 ∈ Ring))
32	31	anim2d 612	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)))
33	32	anim1d 611	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0))))
34	33	imdistanri 569	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0))
35		ovif 7490	. . . . . . . 8 ⊢ (if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)))
36	4	matring 22337	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
37	17	ply1sca 22144	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
39	38	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
40	39	fveq2d 6865	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (0g‘𝐴) = (0g‘(Scalar‘𝑄)))
41	40	oveq1d 7405	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
42	17	ply1lmod 22143	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
43	36, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
44	43	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
45		eqid 2730	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
46		eqid 2730	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘𝑄)
47	45, 46	mgpbas 20061	. . . . . . . . . . . 12 ⊢ (Base‘𝑄) = (Base‘(mulGrp‘𝑄))
48	17	ply1ring 22139	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
49	36, 48	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
50	45	ringmgp 20155	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd)
51	49, 50	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (mulGrp‘𝑄) ∈ Mnd)
52	51	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑄) ∈ Mnd)
53		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
54	16, 17, 46	vr1cl 22109	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝑋 ∈ (Base‘𝑄))
55	36, 54	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑋 ∈ (Base‘𝑄))
56	55	ad2antrr 726	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑄))
57	47, 15, 52, 53, 56	mulgnn0cld 19034	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
58		eqid 2730	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
59		eqid 2730	. . . . . . . . . . . 12 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
60		eqid 2730	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
61	46, 58, 14, 59, 60	lmod0vs 20808	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
62	44, 57, 61	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
63	41, 62	eqtrd 2765	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
64	63	ifeq2d 4512	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋))) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
65	35, 64	eqtrid 2777	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
66	34, 65	syl 17	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
67	30, 66	eqtrd 2765	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
68	67	mpteq2dva 5203	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄))))
69	68	oveq2d 7406	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))))
70		ringmnd 20159	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
71	49, 70	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
72	71	adantr 480	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑄 ∈ Mnd)
73		nn0ex 12455	. . . . . 6 ⊢ ℕ0 ∈ V
74	73	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ℕ0 ∈ V)
75		simprr 772	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝐿 ∈ ℕ0)
76		eqid 2730	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
77	38	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝐴) = (Base‘(Scalar‘𝑄)))
78	5, 77	eqtrid 2777	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐾 = (Base‘(Scalar‘𝑄)))
79	78	eleq2d 2815	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐾 ↔ 𝑀 ∈ (Base‘(Scalar‘𝑄))))
80	79	biimpcd 249	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐾 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ (Base‘(Scalar‘𝑄))))
81	80	adantr 480	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ (Base‘(Scalar‘𝑄))))
82	81	impcom 407	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑀 ∈ (Base‘(Scalar‘𝑄)))
83	82	adantr 480	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ (Base‘(Scalar‘𝑄)))
84		eqid 2730	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
85	46, 58, 14, 84	lmodvscl 20791	. . . . . . 7 ⊢ ((𝑄 ∈ LMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑄)) ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
86	44, 83, 57, 85	syl3anc 1373	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
87	86	ralrimiva 3126	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ∀𝑘 ∈ ℕ0 (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
88	60, 72, 74, 75, 76, 87	gsummpt1n0 19902	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))) = ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)))
89	1, 88	sylanl2 681	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))) = ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)))
90	69, 89	eqtrd 2765	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((((𝐿𝐸𝑌) · (𝑇‘𝑀)) decompPMat 𝑘) ∗ (𝑘 ↑ 𝑋)))) = ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)))
91		csbov2g 7438	. . . 4 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋)))
92		csbov1g 7437	. . . . . 6 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋𝐿 / 𝑘⦌𝑘 ↑ 𝑋))
93		csbvarg 4400	. . . . . . 7 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌𝑘 = 𝐿)
94	93	oveq1d 7405	. . . . . 6 ⊢ (𝐿 ∈ ℕ0 → (⦋𝐿 / 𝑘⦌𝑘 ↑ 𝑋) = (𝐿 ↑ 𝑋))
95	92, 94	eqtrd 2765	. . . . 5 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋) = (𝐿 ↑ 𝑋))
96	95	oveq2d 7406	. . . 4 ⊢ (𝐿 ∈ ℕ0 → (𝑀 ∗ ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
97	91, 96	eqtrd 2765	. . 3 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
98	97	ad2antll 729	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
99	21, 90, 98	3eqtrd 2769	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⦋csb 3865  ifcif 4491   ↦ cmpt 5191  ‘cfv 6514  (class class class)co 7390  Fincfn 8921  ℕ₀cn0 12449  Basecbs 17186  Scalarcsca 17230   ·_𝑠 cvsca 17231  0_gc0g 17409   Σ_g cgsu 17410  Mndcmnd 18668  ._gcmg 19006  mulGrpcmgp 20056  Ringcrg 20149  CRingccrg 20150  LModclmod 20773  var₁cv1 22067  Poly₁cpl1 22068   Mat cmat 22301   matToPolyMat cmat2pmat 22598   decompPMat cdecpmat 22656   pMatToMatPoly cpm2mp 22686

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656  df-ofr 7657  df-om 7846  df-1st 7971  df-2nd 7972  df-supp 8143  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-pm 8805  df-ixp 8874  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-fsupp 9320  df-sup 9400  df-oi 9470  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-fzo 13623  df-seq 13974  df-hash 14303  df-struct 17124  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-mulr 17241  df-sca 17243  df-vsca 17244  df-ip 17245  df-tset 17246  df-ple 17247  df-ds 17249  df-hom 17251  df-cco 17252  df-0g 17411  df-gsum 17412  df-prds 17417  df-pws 17419  df-mre 17554  df-mrc 17555  df-acs 17557  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-mhm 18717  df-submnd 18718  df-grp 18875  df-minusg 18876  df-sbg 18877  df-mulg 19007  df-subg 19062  df-ghm 19152  df-cntz 19256  df-cmn 19719  df-abl 19720  df-mgp 20057  df-rng 20069  df-ur 20098  df-ring 20151  df-cring 20152  df-subrng 20462  df-subrg 20486  df-lmod 20775  df-lss 20845  df-sra 21087  df-rgmod 21088  df-dsmm 21648  df-frlm 21663  df-assa 21769  df-ascl 21771  df-psr 21825  df-mvr 21826  df-mpl 21827  df-opsr 21829  df-psr1 22071  df-vr1 22072  df-ply1 22073  df-coe1 22074  df-mamu 22285  df-mat 22302  df-mat2pmat 22601  df-decpmat 22657  df-pm2mp 22687

This theorem is referenced by:  pm2mp  22719