Theorem pmatcollpwlem 22898

Description: Lemma for pmatcollpw 22899. (Contributed by AV, 26-Oct-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
pmatcollpw.p	⊢ 𝑃 = (Poly1‘𝑅)
pmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw.b	⊢ 𝐵 = (Base‘𝐶)
pmatcollpw.m	⊢ ∗ = ( ·𝑠 ‘𝐶)
pmatcollpw.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
pmatcollpw.x	⊢ 𝑋 = (var1‘𝑅)
pmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

Assertion

Ref	Expression
pmatcollpwlem	⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Proof of Theorem pmatcollpwlem

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmatcollpw.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
2	1	ply1assa 22319	. . . . . . 7 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
3	2	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ AssAlg)
4	3	adantr 485	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ AssAlg)
5	4	3ad2ant1 1149	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑃 ∈ AssAlg)
6		eqid 2765	. . . . . 6 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
7		eqid 2765	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
8		eqid 2765	. . . . . 6 ⊢ (Base‘(𝑁 Mat 𝑅)) = (Base‘(𝑁 Mat 𝑅))
9		simp2 1153	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
10		simp3 1154	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
11		simp2 1153	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ CRing)
12	11	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ CRing)
13		simp3 1154	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
14	13	adantr 485	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ 𝐵)
15		simpr 489	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
16		pmatcollpw.c	. . . . . . . . 9 ⊢ 𝐶 = (𝑁 Mat 𝑃)
17		pmatcollpw.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
18	1, 16, 17, 6, 8	decpmatcl 22885	. . . . . . . 8 ⊢ ((𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵 ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
19	12, 14, 15, 18	syl3anc 1394	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
20	19	3ad2ant1 1149	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
21	6, 7, 8, 9, 10, 20	matecld 22544	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅))
22		crngring 20318	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23	22	3ad2ant2 1150	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ Ring)
24	1	ply1sca 22372	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
25	23, 24	syl 18	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑅 = (Scalar‘𝑃))
26	25	eqcomd 2771	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Scalar‘𝑃) = 𝑅)
27	26	fveq2d 6875	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
28	27	eleq2d 2851	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
29	28	adantr 485	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
30	29	3ad2ant1 1149	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ↔ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘𝑅)))
31	21, 30	mpbird 260	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)))
32		pmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
33		eqid 2765	. . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
34		pmatcollpw.e	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
35		eqid 2765	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
36	1, 32, 33, 34, 35	ply1moncl 22392	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
37	23, 36	sylan 591	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
38	37	3ad2ant1 1149	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑛 ↑ 𝑋) ∈ (Base‘𝑃))
39		eqid 2765	. . . . 5 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
40		eqid 2765	. . . . 5 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2765	. . . . 5 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
42		eqid 2765	. . . . 5 ⊢ (.r‘𝑃) = (.r‘𝑃)
43		eqid 2765	. . . . 5 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
44	39, 40, 41, 35, 42, 43	asclmul2 21997	. . . 4 ⊢ ((𝑃 ∈ AssAlg ∧ (𝑎(𝑀 decompPMat 𝑛)𝑏) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑛 ↑ 𝑋) ∈ (Base‘𝑃)) → ((𝑛 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
45	5, 31, 38, 44	syl3anc 1394	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)))
46		eqidd 2766	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
47		oveq12 7409	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑎(𝑀 decompPMat 𝑛)𝑏))
48	47	fveq2d 6875	. . . . . . 7 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
49	48	adantl 486	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) ∧ (𝑖 = 𝑎 ∧ 𝑗 = 𝑏)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
50		fvexd 6886	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) ∈ V)
51	46, 49, 9, 10, 50	ovmpod 7552	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏) = ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)))
52	51	eqcomd 2771	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏)) = (𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏))
53	52	oveq2d 7416	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑎(𝑀 decompPMat 𝑛)𝑏))) = ((𝑛 ↑ 𝑋)(.r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
54	45, 53	eqtr3d 2802	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = ((𝑛 ↑ 𝑋)(.r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
55	1	ply1ring 22367	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
56	22, 55	syl 18	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
57	56	3ad2ant2 1150	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) → 𝑃 ∈ Ring)
58	57	adantr 485	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑃 ∈ Ring)
59	58	3ad2ant1 1149	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑃 ∈ Ring)
60		simpl1 1208	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
61	12, 22	syl 18	. . . . . . . 8 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
62	61	3ad2ant1 1149	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
63		simp2 1153	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
64		simp3 1154	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
65	19	3ad2ant1 1149	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅)))
66	6, 7, 8, 63, 64, 65	matecld 22544	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅))
67	1, 39, 7, 35	ply1sclcl 22407	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖(𝑀 decompPMat 𝑛)𝑗) ∈ (Base‘𝑅)) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
68	62, 66, 67	syl2anc 595	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)) ∈ (Base‘𝑃))
69	16, 35, 17, 60, 58, 68	matbas2d 22541	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵)
70	37, 69	jca 520	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
71	70	3ad2ant1 1149	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵))
72	9, 10	jca 520	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁))
73		pmatcollpw.m	. . . 4 ⊢ ∗ = ( ·𝑠 ‘𝐶)
74	16, 17, 35, 73, 42	matvscacell 22554	. . 3 ⊢ ((𝑃 ∈ Ring ∧ ((𝑛 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) ∈ 𝐵) ∧ (𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁)) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 ↑ 𝑋)(.r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
75	59, 71, 72, 74	syl3anc 1394	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = ((𝑛 ↑ 𝑋)(.r‘𝑃)(𝑎(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))𝑏)))
76	23	adantr 485	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
77		pmatcollpw.t	. . . . . . . 8 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
78	77, 6, 8, 1, 39	mat2pmatval 22842	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑀 decompPMat 𝑛) ∈ (Base‘(𝑁 Mat 𝑅))) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
79	60, 76, 19, 78	syl3anc 1394	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑇‘(𝑀 decompPMat 𝑛)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))
80	79	eqcomd 2771	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))) = (𝑇‘(𝑀 decompPMat 𝑛)))
81	80	oveq2d 7416	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗)))) = ((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛))))
82	81	oveqd 7417	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
83	82	3ad2ant1 1149	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝑎((𝑛 ↑ 𝑋) ∗ (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((algSc‘𝑃)‘(𝑖(𝑀 decompPMat 𝑛)𝑗))))𝑏) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))
84	54, 75, 83	3eqtr2d 2806	1 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐵) ∧ 𝑛 ∈ ℕ0) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝑎(𝑀 decompPMat 𝑛)𝑏)( ·𝑠 ‘𝑃)(𝑛 ↑ 𝑋)) = (𝑎((𝑛 ↑ 𝑋) ∗ (𝑇‘(𝑀 decompPMat 𝑛)))𝑏))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Fincfn 8931  ℕ₀cn0 12495  Basecbs 17259  ._rcmulr 17301  Scalarcsca 17303   ·_𝑠 cvsca 17304  ._gcmg 19124  mulGrpcmgp 20207  Ringcrg 20306  CRingccrg 20307  AssAlgcasa 21960  algSccascl 21962  var₁cv1 22296  Poly₁cpl1 22297   Mat cmat 22525   matToPolyMat cmat2pmat 22822   decompPMat cdecpmat 22880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-of 7664  df-ofr 7665  df-om 7851  df-1st 7974  df-2nd 7975  df-supp 8145  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-er 8682  df-map 8814  df-pm 8815  df-ixp 8884  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-fsupp 9310  df-sup 9390  df-oi 9460  df-card 9913  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-7 12299  df-8 12300  df-9 12301  df-n0 12496  df-z 12583  df-dec 12703  df-uz 12854  df-fz 13527  df-fzo 13674  df-seq 14029  df-hash 14358  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-ip 17318  df-tset 17319  df-ple 17320  df-ds 17322  df-hom 17324  df-cco 17325  df-0g 17484  df-gsum 17485  df-prds 17490  df-pws 17492  df-mre 17628  df-mrc 17629  df-acs 17631  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-mhm 18831  df-submnd 18832  df-grp 18993  df-minusg 18994  df-sbg 18995  df-mulg 19125  df-subg 19180  df-ghm 19275  df-cntz 19378  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-cring 20309  df-subrng 20622  df-subrg 20646  df-lmod 20952  df-lss 21022  df-sra 21263  df-rgmod 21264  df-dsmm 21842  df-frlm 21857  df-assa 21963  df-ascl 21965  df-psr 22019  df-mvr 22020  df-mpl 22021  df-opsr 22023  df-psr1 22300  df-vr1 22301  df-ply1 22302  df-coe1 22303  df-mat 22526  df-mat2pmat 22825  df-decpmat 22881

This theorem is referenced by:  pmatcollpw  22899