Theorem monmat2matmon 21426

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 19302	. . 3 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2		simpll 765	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑁 ∈ Fin)
3		simplr 767	. . . 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 21342	. . . 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 21398	. . . 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 1367	. . 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 679	. 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 765	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
23		simpr 487	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0))
24	23	anim1i 616	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0))
25		df-3an 1085	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0) ↔ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) ∧ 𝑘 ∈ ℕ0))
26	24, 25	sylibr 236	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0))
27		eqid 2821	. . . . . . . . 9 ⊢ (0g‘𝐴) = (0g‘𝐴)
28	7, 8, 4, 5, 27, 11, 12, 10, 6	monmatcollpw 21381	. . . . . . . 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 586	. . . . . . 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 7165	. . . . . 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 613	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring)))
33	32	anim1d 612	. . . . . . . 8 ⊢ (𝑘 ∈ ℕ0 → (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0))))
34	33	imdistanri 572	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0))
35		ovif 7245	. . . . . . . 8 ⊢ (if(𝑘 = 𝐿, 𝑀, (0g‘𝐴)) ∗ (𝑘 ↑ 𝑋)) = if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)))
36	4	matring 21046	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
37	17	ply1sca 20415	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
39	38	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝐴 = (Scalar‘𝑄))
40	39	fveq2d 6668	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (0g‘𝐴) = (0g‘(Scalar‘𝑄)))
41	40	oveq1d 7165	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)))
42	17	ply1lmod 20414	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ LMod)
43	36, 42	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
44	43	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑄 ∈ LMod)
45	17	ply1ring 20410	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ring → 𝑄 ∈ Ring)
46	36, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Ring)
47		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑄) = (mulGrp‘𝑄)
48	47	ringmgp 19297	. . . . . . . . . . . . . 14 ⊢ (𝑄 ∈ Ring → (mulGrp‘𝑄) ∈ Mnd)
49	46, 48	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (mulGrp‘𝑄) ∈ Mnd)
50	49	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (mulGrp‘𝑄) ∈ Mnd)
51		simpr 487	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0)
52		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑄) = (Base‘𝑄)
53	16, 17, 52	vr1cl 20379	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ Ring → 𝑋 ∈ (Base‘𝑄))
54	36, 53	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑋 ∈ (Base‘𝑄))
55	54	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑋 ∈ (Base‘𝑄))
56	47, 52	mgpbas 19239	. . . . . . . . . . . . 13 ⊢ (Base‘𝑄) = (Base‘(mulGrp‘𝑄))
57	56, 15	mulgnn0cl 18238	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑄) ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ 𝑋 ∈ (Base‘𝑄)) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
58	50, 51, 55, 57	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑘 ↑ 𝑋) ∈ (Base‘𝑄))
59		eqid 2821	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑄) = (Scalar‘𝑄)
60		eqid 2821	. . . . . . . . . . . 12 ⊢ (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
61		eqid 2821	. . . . . . . . . . . 12 ⊢ (0g‘𝑄) = (0g‘𝑄)
62	52, 59, 14, 60, 61	lmod0vs 19661	. . . . . . . . . . 11 ⊢ ((𝑄 ∈ LMod ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
63	44, 58, 62	syl2anc 586	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
64	41, 63	eqtrd 2856	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → ((0g‘𝐴) ∗ (𝑘 ↑ 𝑋)) = (0g‘𝑄))
65	64	ifeq2d 4485	. . . . . . . 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)))
66	35, 65	syl5eq 2868	. . . . . . 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)))
67	34, 66	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)))
68	30, 67	eqtrd 2856	. . . . 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)))
69	68	mpteq2dva 5153	. . . 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))))
70	69	oveq2d 7166	. . 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)))))
71		ringmnd 19300	. . . . . . 7 ⊢ (𝑄 ∈ Ring → 𝑄 ∈ Mnd)
72	46, 71	syl 17	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ Mnd)
73	72	adantr 483	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑄 ∈ Mnd)
74		nn0ex 11897	. . . . . 6 ⊢ ℕ0 ∈ V
75	74	a1i 11	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ℕ0 ∈ V)
76		simprr 771	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝐿 ∈ ℕ0)
77		eqid 2821	. . . . 5 ⊢ (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 𝐿, (𝑀 ∗ (𝑘 ↑ 𝑋)), (0g‘𝑄)))
78	38	fveq2d 6668	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝐴) = (Base‘(Scalar‘𝑄)))
79	5, 78	syl5eq 2868	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐾 = (Base‘(Scalar‘𝑄)))
80	79	eleq2d 2898	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀 ∈ 𝐾 ↔ 𝑀 ∈ (Base‘(Scalar‘𝑄))))
81	80	biimpcd 251	. . . . . . . . . 10 ⊢ (𝑀 ∈ 𝐾 → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ (Base‘(Scalar‘𝑄))))
82	81	adantr 483	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑀 ∈ (Base‘(Scalar‘𝑄))))
83	82	impcom 410	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → 𝑀 ∈ (Base‘(Scalar‘𝑄)))
84	83	adantr 483	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → 𝑀 ∈ (Base‘(Scalar‘𝑄)))
85		eqid 2821	. . . . . . . 8 ⊢ (Base‘(Scalar‘𝑄)) = (Base‘(Scalar‘𝑄))
86	52, 59, 14, 85	lmodvscl 19645	. . . . . . 7 ⊢ ((𝑄 ∈ LMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑄)) ∧ (𝑘 ↑ 𝑋) ∈ (Base‘𝑄)) → (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
87	44, 84, 58, 86	syl3anc 1367	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) ∧ 𝑘 ∈ ℕ0) → (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
88	87	ralrimiva 3182	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ∀𝑘 ∈ ℕ0 (𝑀 ∗ (𝑘 ↑ 𝑋)) ∈ (Base‘𝑄))
89	61, 73, 75, 76, 77, 88	gsummpt1n0 19079	. . . 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)))
90	1, 89	sylanl2 679	. . 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)))
91	70, 90	eqtrd 2856	. 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)))
92		csbov2g 7196	. . . 4 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋)))
93		csbov1g 7195	. . . . . 6 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋) = (⦋𝐿 / 𝑘⦌𝑘 ↑ 𝑋))
94		csbvarg 4382	. . . . . . 7 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌𝑘 = 𝐿)
95	94	oveq1d 7165	. . . . . 6 ⊢ (𝐿 ∈ ℕ0 → (⦋𝐿 / 𝑘⦌𝑘 ↑ 𝑋) = (𝐿 ↑ 𝑋))
96	93, 95	eqtrd 2856	. . . . 5 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋) = (𝐿 ↑ 𝑋))
97	96	oveq2d 7166	. . . 4 ⊢ (𝐿 ∈ ℕ0 → (𝑀 ∗ ⦋𝐿 / 𝑘⦌(𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
98	92, 97	eqtrd 2856	. . 3 ⊢ (𝐿 ∈ ℕ0 → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
99	98	ad2antll 727	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → ⦋𝐿 / 𝑘⦌(𝑀 ∗ (𝑘 ↑ 𝑋)) = (𝑀 ∗ (𝐿 ↑ 𝑋)))
100	21, 91, 99	3eqtrd 2860	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0)) → (𝐼‘((𝐿𝐸𝑌) · (𝑇‘𝑀))) = (𝑀 ∗ (𝐿 ↑ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⦋csb 3882  ifcif 4466   ↦ cmpt 5138  ‘cfv 6349  (class class class)co 7150  Fincfn 8503  ℕ₀cn0 11891  Basecbs 16477  Scalarcsca 16562   ·_𝑠 cvsca 16563  0_gc0g 16707   Σ_g cgsu 16708  Mndcmnd 17905  ._gcmg 18218  mulGrpcmgp 19233  Ringcrg 19291  CRingccrg 19292  LModclmod 19628  var₁cv1 20338  Poly₁cpl1 20339   Mat cmat 21010   matToPolyMat cmat2pmat 21306   decompPMat cdecpmat 21364   pMatToMatPoly cpm2mp 21394

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-ot 4569  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-ofr 7404  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-map 8402  df-pm 8403  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-sup 8900  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-dec 12093  df-uz 12238  df-fz 12887  df-fzo 13028  df-seq 13364  df-hash 13685  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-ip 16577  df-tset 16578  df-ple 16579  df-ds 16581  df-hom 16583  df-cco 16584  df-0g 16709  df-gsum 16710  df-prds 16715  df-pws 16717  df-mre 16851  df-mrc 16852  df-acs 16854  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-mhm 17950  df-submnd 17951  df-grp 18100  df-minusg 18101  df-sbg 18102  df-mulg 18219  df-subg 18270  df-ghm 18350  df-cntz 18441  df-cmn 18902  df-abl 18903  df-mgp 19234  df-ur 19246  df-ring 19293  df-cring 19294  df-subrg 19527  df-lmod 19630  df-lss 19698  df-sra 19938  df-rgmod 19939  df-assa 20079  df-ascl 20081  df-psr 20130  df-mvr 20131  df-mpl 20132  df-opsr 20134  df-psr1 20342  df-vr1 20343  df-ply1 20344  df-coe1 20345  df-dsmm 20870  df-frlm 20885  df-mamu 20989  df-mat 21011  df-mat2pmat 21309  df-decpmat 21365  df-pm2mp 21395

This theorem is referenced by:  pm2mp  21427