Theorem monmatcollpw 22128

Description: The matrix consisting of the coefficients in the polynomial entries of a polynomial matrix having scaled monomials with the same power as entries is the matrix of the coefficients of the monomials or a zero matrix. Generalization of decpmatid 22119 (but requires 𝑅 to be commutative!). (Contributed by AV, 11-Nov-2019.) (Revised by AV, 4-Dec-2019.)

Hypotheses

Ref	Expression
monmatcollpw.p	⊢ 𝑃 = (Poly1‘𝑅)
monmatcollpw.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
monmatcollpw.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
monmatcollpw.k	⊢ 𝐾 = (Base‘𝐴)
monmatcollpw.0	⊢ 0 = (0g‘𝐴)
monmatcollpw.e	⊢ ↑ = (.g‘(mulGrp‘𝑃))
monmatcollpw.x	⊢ 𝑋 = (var1‘𝑅)
monmatcollpw.m	⊢ · = ( ·𝑠 ‘𝐶)
monmatcollpw.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)

Assertion

Ref	Expression
monmatcollpw	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))

Proof of Theorem monmatcollpw

Dummy variables 𝑖 𝑗 𝑙 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll 765	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2		crngring 19976	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		monmatcollpw.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 21619	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6	5	ad2antlr 725	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
7	2	adantl 482	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8		simp2 1137	. . . . . 6 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9		monmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
10		eqid 2736	. . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
11		monmatcollpw.e	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
12		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
13	3, 9, 10, 11, 12	ply1moncl 21642	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
14	7, 8, 13	syl2an 596	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
15	2	anim2i 617	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16		simp1 1136	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ 𝐾)
17	15, 16	anim12i 613	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾))
18		df-3an 1089	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾))
19	17, 18	sylibr 233	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾))
20		monmatcollpw.t	. . . . . . 7 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
21		monmatcollpw.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
22		monmatcollpw.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐴)
23		monmatcollpw.c	. . . . . . 7 ⊢ 𝐶 = (𝑁 Mat 𝑃)
24	20, 21, 22, 3, 23	mat2pmatbas 22075	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) → (𝑇‘𝑀) ∈ (Base‘𝐶))
25	19, 24	syl 17	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑇‘𝑀) ∈ (Base‘𝐶))
26	14, 25	jca 512	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶)))
27		eqid 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		monmatcollpw.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝐶)
29	12, 23, 27, 28	matvscl 21780	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) ∧ ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶))) → ((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) ∈ (Base‘𝐶))
30	1, 6, 26, 29	syl21anc 836	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) ∈ (Base‘𝐶))
31		simpr3 1196	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
32	23, 27	decpmatval 22114	. . 3 ⊢ ((((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) ∈ (Base‘𝐶) ∧ 𝐼 ∈ ℕ0) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗))‘𝐼)))
33	30, 31, 32	syl2anc 584	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗))‘𝐼)))
34	6	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ Ring)
35	26	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶)))
36		3simpc 1150	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
37		eqid 2736	. . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃)
38	23, 27, 12, 28, 37	matvscacell 21785	. . . . . . 7 ⊢ ((𝑃 ∈ Ring ∧ ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶)) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗) = ((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)))
39	34, 35, 36, 38	syl3anc 1371	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗) = ((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)))
40	39	fveq2d 6846	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗)) = (coe1‘((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗))))
41	40	fveq1d 6844	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗))‘𝐼) = ((coe1‘((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)))‘𝐼))
42	16	anim2i 617	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ 𝐾))
43		df-3an 1089	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ 𝐾))
44	42, 43	sylibr 233	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾))
45	44	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾))
46		eqid 2736	. . . . . . . . . 10 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
47	20, 21, 22, 3, 46	mat2pmatvalel 22074	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
48	45, 36, 47	syl2anc 584	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖(𝑇‘𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
49	48	oveq2d 7373	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)) = ((𝐿 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))))
50	3	ply1assa 21570	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
51	50	ad2antlr 725	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52	51	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ AssAlg)
53		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘𝐴) = (Base‘𝐴)
55		simp2 1137	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
56		simp3 1138	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
57	22	eleq2i 2829	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ 𝐾 ↔ 𝑀 ∈ (Base‘𝐴))
58	57	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐾 → 𝑀 ∈ (Base‘𝐴))
59	58	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
60	59	adantl 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61	60	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
62	21, 53, 54, 55, 56, 61	matecld 21775	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
63	3	ply1sca 21624	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
64	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
65	64	eqcomd 2742	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
66	65	fveq2d 6846	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
67	66	adantr 481	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
68	67	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
69	62, 68	eleqtrrd 2841	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70	14	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
71		eqid 2736	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
72		eqid 2736	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73		eqid 2736	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
74	46, 71, 72, 12, 37, 73	asclmul2 21290	. . . . . . . 8 ⊢ ((𝑃 ∈ AssAlg ∧ (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝐿 ↑ 𝑋) ∈ (Base‘𝑃)) → ((𝐿 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋)))
75	52, 69, 70, 74	syl3anc 1371	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝐿 ↑ 𝑋)(.r‘𝑃)((algSc‘𝑃)‘(𝑖𝑀𝑗))) = ((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋)))
76	49, 75	eqtrd 2776	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)) = ((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋)))
77	76	fveq2d 6846	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗))) = (coe1‘((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋))))
78	77	fveq1d 6844	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘((𝐿 ↑ 𝑋)(.r‘𝑃)(𝑖(𝑇‘𝑀)𝑗)))‘𝐼) = ((coe1‘((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋)))‘𝐼))
79	2	ad2antlr 725	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑅 ∈ Ring)
80	79	3ad2ant1 1133	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
81		simp1r2 1270	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ ℕ0)
82		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
83	82, 53, 3, 9, 73, 10, 11	coe1tm 21644	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑀𝑗) ∈ (Base‘𝑅) ∧ 𝐿 ∈ ℕ0) → (coe1‘((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))))
84	80, 62, 81, 83	syl3anc 1371	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))))
85	84	fveq1d 6844	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘((𝑖𝑀𝑗)( ·𝑠 ‘𝑃)(𝐿 ↑ 𝑋)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))
86	41, 78, 85	3eqtrd 2780	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))
87	86	mpoeq3dva 7434	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖((𝐿 ↑ 𝑋) · (𝑇‘𝑀))𝑗))‘𝐼)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)))
88		monmatcollpw.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐴)
89	15	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
90	89	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
91	21, 82	mat0op 21768	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑧 ∈ 𝑁, 𝑤 ∈ 𝑁 ↦ (0g‘𝑅)))
92	90, 91	syl 17	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0g‘𝐴) = (𝑧 ∈ 𝑁, 𝑤 ∈ 𝑁 ↦ (0g‘𝑅)))
93	88, 92	eqtrid 2788	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 0 = (𝑧 ∈ 𝑁, 𝑤 ∈ 𝑁 ↦ (0g‘𝑅)))
94		eqidd 2737	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ (𝑧 = 𝑥 ∧ 𝑤 = 𝑦)) → (0g‘𝑅) = (0g‘𝑅))
95		simprl 769	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
96		simprr 771	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
97		fvexd 6857	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0g‘𝑅) ∈ V)
98	93, 94, 95, 96, 97	ovmpod 7507	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥 0 𝑦) = (0g‘𝑅))
99	98	eqcomd 2742	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0g‘𝑅) = (𝑥 0 𝑦))
100	99	ifeq2d 4506	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
101		eqidd 2737	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)))
102		oveq12 7366	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103	102	ifeq1d 4505	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
104	103	mpteq2dv 5207	. . . . . . . 8 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅))))
105	104	fveq1d 6844	. . . . . . 7 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼) = ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))‘𝐼))
106		eqidd 2737	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅))))
107		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑙 = 𝐼 → (𝑙 = 𝐿 ↔ 𝐼 = 𝐿))
108	107	ifbid 4509	. . . . . . . . 9 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
109	108	adantl 482	. . . . . . . 8 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
110	31	adantr 481	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐼 ∈ ℕ0)
111		ovex 7390	. . . . . . . . . 10 ⊢ (𝑥𝑀𝑦) ∈ V
112		fvex 6855	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
113	111, 112	ifex 4536	. . . . . . . . 9 ⊢ if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) ∈ V
114	113	a1i 11	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) ∈ V)
115	106, 109, 110, 114	fvmptd 6955	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
116	105, 115	sylan9eqr 2798	. . . . . 6 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) ∧ (𝑖 = 𝑥 ∧ 𝑗 = 𝑦)) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
117	101, 116, 95, 96, 114	ovmpod 7507	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
118		ifov 7457	. . . . . 6 ⊢ (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦))
119	118	a1i 11	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (𝑥 0 𝑦)))
120	100, 117, 119	3eqtr4d 2786	. . . 4 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦))
121	120	ralrimivva 3197	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦))
122		simplr 767	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123		eqidd 2737	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))))
124	107	ifbid 4509	. . . . . . . 8 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))
125	124	adantl 482	. . . . . . 7 ⊢ (((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) ∧ 𝑙 = 𝐼) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))
126	31	3ad2ant1 1133	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐼 ∈ ℕ0)
127	53, 82	ring0cl 19990	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
128	7, 127	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g‘𝑅) ∈ (Base‘𝑅))
129	128	adantr 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (0g‘𝑅) ∈ (Base‘𝑅))
130	129	3ad2ant1 1133	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (0g‘𝑅) ∈ (Base‘𝑅))
131	62, 130	ifcld 4532	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) ∈ (Base‘𝑅))
132	123, 125, 126, 131	fvmptd 6955	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))
133	132, 131	eqeltrd 2838	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼) ∈ (Base‘𝑅))
134	21, 53, 22, 1, 122, 133	matbas2d 21772	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) ∈ 𝐾)
135	60, 57	sylibr 233	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑀 ∈ 𝐾)
136	21	matring 21792	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
137	22, 88	ring0cl 19990	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐾)
138	15, 136, 137	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0 ∈ 𝐾)
139	138	adantr 481	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 0 ∈ 𝐾)
140	135, 139	ifcld 4532	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
141	21, 22	eqmat 21773	. . . 4 ⊢ (((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) ∈ 𝐾 ∧ if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦)))
142	134, 140, 141	syl2anc 584	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ) ↔ ∀𝑥 ∈ 𝑁 ∀𝑦 ∈ 𝑁 (𝑥(𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼))𝑦) = (𝑥if(𝐼 = 𝐿, 𝑀, 0 )𝑦)))
143	121, 142	mpbird 256	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))‘𝐼)) = if(𝐼 = 𝐿, 𝑀, 0 ))
144	33, 87, 143	3eqtrd 2780	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) decompPMat 𝐼) = if(𝐼 = 𝐿, 𝑀, 0 ))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  Vcvv 3445  ifcif 4486   ↦ cmpt 5188  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  Fincfn 8883  ℕ₀cn0 12413  Basecbs 17083  ._rcmulr 17134  Scalarcsca 17136   ·_𝑠 cvsca 17137  0_gc0g 17321  ._gcmg 18872  mulGrpcmgp 19896  Ringcrg 19964  CRingccrg 19965  AssAlgcasa 21256  algSccascl 21258  var₁cv1 21547  Poly₁cpl1 21548  coe₁cco1 21549   Mat cmat 21754   matToPolyMat cmat2pmat 22053   decompPMat cdecpmat 22111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-assa 21259  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mamu 21733  df-mat 21755  df-mat2pmat 22056  df-decpmat 22112

This theorem is referenced by:  monmat2matmon  22173