Theorem monmatcollpw 21384

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 21375 (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 766	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑁 ∈ Fin)
2		crngring 19302	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		monmatcollpw.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 20877	. . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
5	2, 4	syl 17	. . . . 5 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ Ring)
6	5	ad2antlr 726	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑃 ∈ Ring)
7	2	adantl 485	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8		simp2 1134	. . . . . 6 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9		monmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
10		eqid 2798	. . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
11		monmatcollpw.e	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
12		eqid 2798	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
13	3, 9, 10, 11, 12	ply1moncl 20900	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
14	7, 8, 13	syl2an 598	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
15	2	anim2i 619	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16		simp1 1133	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ 𝐾)
17	15, 16	anim12i 615	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾))
18		df-3an 1086	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾))
19	17, 18	sylibr 237	. . . . . 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 21331	. . . . . 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 515	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶)))
27		eqid 2798	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		monmatcollpw.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝐶)
29	12, 23, 27, 28	matvscl 21036	. . . 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 1193	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
32	23, 27	decpmatval 21370	. . 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 587	. 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 1130	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ Ring)
35	26	3ad2ant1 1130	. . . . . . 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 1147	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
37		eqid 2798	. . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃)
38	23, 27, 12, 28, 37	matvscacell 21041	. . . . . . 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 1368	. . . . . 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 6649	. . . . 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 6647	. . . 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 619	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ 𝐾))
43		df-3an 1086	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) ↔ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ 𝐾))
44	42, 43	sylibr 237	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾))
45	44	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾))
46		eqid 2798	. . . . . . . . . 10 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
47	20, 21, 22, 3, 46	mat2pmatvalel 21330	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
48	45, 36, 47	syl2anc 587	. . . . . . . 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 7151	. . . . . . 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 20828	. . . . . . . . . 10 ⊢ (𝑅 ∈ CRing → 𝑃 ∈ AssAlg)
51	50	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑃 ∈ AssAlg)
52	51	3ad2ant1 1130	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ AssAlg)
53		eqid 2798	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2798	. . . . . . . . . 10 ⊢ (Base‘𝐴) = (Base‘𝐴)
55		simp2 1134	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
56		simp3 1135	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
57	22	eleq2i 2881	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ 𝐾 ↔ 𝑀 ∈ (Base‘𝐴))
58	57	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐾 → 𝑀 ∈ (Base‘𝐴))
59	58	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
60	59	adantl 485	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61	60	3ad2ant1 1130	. . . . . . . . . 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 21031	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
63	3	ply1sca 20882	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
64	63	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
65	64	eqcomd 2804	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
66	65	fveq2d 6649	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
67	66	adantr 484	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
68	67	3ad2ant1 1130	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
69	62, 68	eleqtrrd 2893	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70	14	3ad2ant1 1130	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
71		eqid 2798	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
72		eqid 2798	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73		eqid 2798	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
74	46, 71, 72, 12, 37, 73	asclmul2 20572	. . . . . . . 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 1368	. . . . . . 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 2833	. . . . . 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 6649	. . . . 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 6647	. . . 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 726	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑅 ∈ Ring)
80	79	3ad2ant1 1130	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
81		simp1r2 1267	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ ℕ0)
82		eqid 2798	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
83	82, 53, 3, 9, 73, 10, 11	coe1tm 20902	. . . . . 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 1368	. . . . 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 6647	. . . 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 2837	. . 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 7210	. 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 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
90	89	adantr 484	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
91	21, 82	mat0op 21024	. . . . . . . . . 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	syl5eq 2845	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 0 = (𝑧 ∈ 𝑁, 𝑤 ∈ 𝑁 ↦ (0g‘𝑅)))
94		eqidd 2799	. . . . . . . 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 770	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑥 ∈ 𝑁)
96		simprr 772	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝑦 ∈ 𝑁)
97		fvexd 6660	. . . . . . . 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 7281	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥 0 𝑦) = (0g‘𝑅))
99	98	eqcomd 2804	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0g‘𝑅) = (𝑥 0 𝑦))
100	99	ifeq2d 4444	. . . . 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 2799	. . . . . 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 7144	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103	102	ifeq1d 4443	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
104	103	mpteq2dv 5126	. . . . . . . 8 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅))))
105	104	fveq1d 6647	. . . . . . 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 2799	. . . . . . . 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 2802	. . . . . . . . . 10 ⊢ (𝑙 = 𝐼 → (𝑙 = 𝐿 ↔ 𝐼 = 𝐿))
108	107	ifbid 4447	. . . . . . . . 9 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
109	108	adantl 485	. . . . . . . 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 484	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐼 ∈ ℕ0)
111		ovex 7168	. . . . . . . . . 10 ⊢ (𝑥𝑀𝑦) ∈ V
112		fvex 6658	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
113	111, 112	ifex 4473	. . . . . . . . 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 6752	. . . . . . 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 2855	. . . . . 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 7281	. . . . 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 7233	. . . . . 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 2843	. . . 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 3156	. . 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 768	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑅 ∈ CRing)
123		eqidd 2799	. . . . . . 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 4447	. . . . . . . 8 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))
125	124	adantl 485	. . . . . . 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 1130	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐼 ∈ ℕ0)
127	53, 82	ring0cl 19315	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
128	7, 127	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g‘𝑅) ∈ (Base‘𝑅))
129	128	adantr 484	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (0g‘𝑅) ∈ (Base‘𝑅))
130	129	3ad2ant1 1130	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (0g‘𝑅) ∈ (Base‘𝑅))
131	62, 130	ifcld 4470	. . . . . . 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 6752	. . . . . 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 2890	. . . . 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 21028	. . . 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 237	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑀 ∈ 𝐾)
136	21	matring 21048	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
137	22, 88	ring0cl 19315	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐾)
138	15, 136, 137	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0 ∈ 𝐾)
139	138	adantr 484	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 0 ∈ 𝐾)
140	135, 139	ifcld 4470	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
141	21, 22	eqmat 21029	. . . 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 587	. . 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 260	. 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 2837	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  Vcvv 3441  ifcif 4425   ↦ cmpt 5110  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Fincfn 8492  ℕ₀cn0 11885  Basecbs 16475  ._rcmulr 16558  Scalarcsca 16560   ·_𝑠 cvsca 16561  0_gc0g 16705  ._gcmg 18216  mulGrpcmgp 19232  Ringcrg 19290  CRingccrg 19291  AssAlgcasa 20539  algSccascl 20541  var₁cv1 20805  Poly₁cpl1 20806  coe₁cco1 20807   Mat cmat 21012   matToPolyMat cmat2pmat 21309   decompPMat cdecpmat 21367

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-ofr 7390  df-om 7561  df-1st 7671  df-2nd 7672  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-fsupp 8818  df-sup 8890  df-oi 8958  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-ip 16575  df-tset 16576  df-ple 16577  df-ds 16579  df-hom 16581  df-cco 16582  df-0g 16707  df-gsum 16708  df-prds 16713  df-pws 16715  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-mulg 18217  df-subg 18268  df-ghm 18348  df-cntz 18439  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-cring 19293  df-subrg 19526  df-lmod 19629  df-lss 19697  df-sra 19937  df-rgmod 19938  df-dsmm 20421  df-frlm 20436  df-assa 20542  df-ascl 20544  df-psr 20594  df-mvr 20595  df-mpl 20596  df-opsr 20598  df-psr1 20809  df-vr1 20810  df-ply1 20811  df-coe1 20812  df-mamu 20991  df-mat 21013  df-mat2pmat 21312  df-decpmat 21368

This theorem is referenced by:  monmat2matmon  21429