Theorem monmatcollpw 22250

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 22241 (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 20050	. . . . . 6 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
3		monmatcollpw.p	. . . . . . 7 ⊢ 𝑃 = (Poly1‘𝑅)
4	3	ply1ring 21741	. . . . . 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 483	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
8		simp2 1138	. . . . . 6 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝐿 ∈ ℕ0)
9		monmatcollpw.x	. . . . . . 7 ⊢ 𝑋 = (var1‘𝑅)
10		eqid 2733	. . . . . . 7 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
11		monmatcollpw.e	. . . . . . 7 ⊢ ↑ = (.g‘(mulGrp‘𝑃))
12		eqid 2733	. . . . . . 7 ⊢ (Base‘𝑃) = (Base‘𝑃)
13	3, 9, 10, 11, 12	ply1moncl 21764	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝐿 ∈ ℕ0) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
14	7, 8, 13	syl2an 597	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
15	2	anim2i 618	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
16		simp1 1137	. . . . . . . 8 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ 𝐾)
17	15, 16	anim12i 614	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ 𝐾))
18		df-3an 1090	. . . . . . 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 22197	. . . . . 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 513	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) ∈ (Base‘𝑃) ∧ (𝑇‘𝑀) ∈ (Base‘𝐶)))
27		eqid 2733	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
28		monmatcollpw.m	. . . . 5 ⊢ · = ( ·𝑠 ‘𝐶)
29	12, 23, 27, 28	matvscl 21902	. . . 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 837	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝐿 ↑ 𝑋) · (𝑇‘𝑀)) ∈ (Base‘𝐶))
31		simpr3 1197	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝐼 ∈ ℕ0)
32	23, 27	decpmatval 22236	. . 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 585	. 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 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ Ring)
35	26	3ad2ant1 1134	. . . . . . 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 1151	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁))
37		eqid 2733	. . . . . . . 8 ⊢ (.r‘𝑃) = (.r‘𝑃)
38	23, 27, 12, 28, 37	matvscacell 21907	. . . . . . 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 1372	. . . . . 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 6885	. . . . 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 6883	. . . 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 618	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ 𝑀 ∈ 𝐾))
43		df-3an 1090	. . . . . . . . . . 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 1134	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾))
46		eqid 2733	. . . . . . . . . 10 ⊢ (algSc‘𝑃) = (algSc‘𝑃)
47	20, 21, 22, 3, 46	mat2pmatvalel 22196	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀 ∈ 𝐾) ∧ (𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁)) → (𝑖(𝑇‘𝑀)𝑗) = ((algSc‘𝑃)‘(𝑖𝑀𝑗)))
48	45, 36, 47	syl2anc 585	. . . . . . . 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 7412	. . . . . . 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 21692	. . . . . . . . . 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 1134	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑃 ∈ AssAlg)
53		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2733	. . . . . . . . . 10 ⊢ (Base‘𝐴) = (Base‘𝐴)
55		simp2 1138	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
56		simp3 1139	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
57	22	eleq2i 2826	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ 𝐾 ↔ 𝑀 ∈ (Base‘𝐴))
58	57	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ 𝐾 → 𝑀 ∈ (Base‘𝐴))
59	58	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0) → 𝑀 ∈ (Base‘𝐴))
60	59	adantl 483	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 𝑀 ∈ (Base‘𝐴))
61	60	3ad2ant1 1134	. . . . . . . . . 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 21897	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
63	3	ply1sca 21746	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ CRing → 𝑅 = (Scalar‘𝑃))
64	63	adantl 483	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 = (Scalar‘𝑃))
65	64	eqcomd 2739	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑃) = 𝑅)
66	65	fveq2d 6885	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
67	66	adantr 482	. . . . . . . . . 10 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
68	67	3ad2ant1 1134	. . . . . . . . 9 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (Base‘(Scalar‘𝑃)) = (Base‘𝑅))
69	62, 68	eleqtrrd 2837	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘(Scalar‘𝑃)))
70	14	3ad2ant1 1134	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝐿 ↑ 𝑋) ∈ (Base‘𝑃))
71		eqid 2733	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
72		eqid 2733	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
73		eqid 2733	. . . . . . . . 9 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
74	46, 71, 72, 12, 37, 73	asclmul2 21412	. . . . . . . 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 1372	. . . . . . 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 2773	. . . . . 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 6885	. . . . 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 6883	. . . 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 1134	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
81		simp1r2 1271	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐿 ∈ ℕ0)
82		eqid 2733	. . . . . . 7 ⊢ (0g‘𝑅) = (0g‘𝑅)
83	82, 53, 3, 9, 73, 10, 11	coe1tm 21766	. . . . . 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 1372	. . . . 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 6883	. . . 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 2777	. . 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 7473	. 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 482	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
90	89	adantr 482	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
91	21, 82	mat0op 21890	. . . . . . . . . 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 2785	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 0 = (𝑧 ∈ 𝑁, 𝑤 ∈ 𝑁 ↦ (0g‘𝑅)))
94		eqidd 2734	. . . . . . . 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 6896	. . . . . . . 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 7547	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (𝑥 0 𝑦) = (0g‘𝑅))
99	98	eqcomd 2739	. . . . . 6 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → (0g‘𝑅) = (𝑥 0 𝑦))
100	99	ifeq2d 4544	. . . . 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 2734	. . . . . 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 7405	. . . . . . . . . 10 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑖𝑀𝑗) = (𝑥𝑀𝑦))
103	102	ifeq1d 4543	. . . . . . . . 9 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
104	103	mpteq2dv 5246	. . . . . . . 8 ⊢ ((𝑖 = 𝑥 ∧ 𝑗 = 𝑦) → (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅))) = (𝑙 ∈ ℕ0 ↦ if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅))))
105	104	fveq1d 6883	. . . . . . 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 2734	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (𝑙 = 𝐼 → (𝑙 = 𝐿 ↔ 𝐼 = 𝐿))
108	107	ifbid 4547	. . . . . . . . 9 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑥𝑀𝑦), (0g‘𝑅)))
109	108	adantl 483	. . . . . . . 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 482	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ (𝑥 ∈ 𝑁 ∧ 𝑦 ∈ 𝑁)) → 𝐼 ∈ ℕ0)
111		ovex 7429	. . . . . . . . . 10 ⊢ (𝑥𝑀𝑦) ∈ V
112		fvex 6894	. . . . . . . . . 10 ⊢ (0g‘𝑅) ∈ V
113	111, 112	ifex 4574	. . . . . . . . 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 6994	. . . . . . 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 2795	. . . . . 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 7547	. . . . 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 7496	. . . . . 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 2783	. . . 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 3201	. . 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 2734	. . . . . . 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 4547	. . . . . . . 8 ⊢ (𝑙 = 𝐼 → if(𝑙 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)) = if(𝐼 = 𝐿, (𝑖𝑀𝑗), (0g‘𝑅)))
125	124	adantl 483	. . . . . . 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 1134	. . . . . . 7 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝐼 ∈ ℕ0)
127	53, 82	ring0cl 20065	. . . . . . . . . . 11 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
128	7, 127	syl 17	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (0g‘𝑅) ∈ (Base‘𝑅))
129	128	adantr 482	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → (0g‘𝑅) ∈ (Base‘𝑅))
130	129	3ad2ant1 1134	. . . . . . . 8 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (0g‘𝑅) ∈ (Base‘𝑅))
131	62, 130	ifcld 4570	. . . . . . 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 6994	. . . . . 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 2834	. . . . 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 21894	. . . 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 21914	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
137	22, 88	ring0cl 20065	. . . . . . 7 ⊢ (𝐴 ∈ Ring → 0 ∈ 𝐾)
138	15, 136, 137	3syl 18	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 0 ∈ 𝐾)
139	138	adantr 482	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → 0 ∈ 𝐾)
140	135, 139	ifcld 4570	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) ∧ (𝑀 ∈ 𝐾 ∧ 𝐿 ∈ ℕ0 ∧ 𝐼 ∈ ℕ0)) → if(𝐼 = 𝐿, 𝑀, 0 ) ∈ 𝐾)
141	21, 22	eqmat 21895	. . . 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 585	. . 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 257	. 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 2777	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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  ifcif 4524   ↦ cmpt 5227  ‘cfv 6535  (class class class)co 7396   ∈ cmpo 7398  Fincfn 8927  ℕ₀cn0 12459  Basecbs 17131  ._rcmulr 17185  Scalarcsca 17187   ·_𝑠 cvsca 17188  0_gc0g 17372  ._gcmg 18935  mulGrpcmgp 19970  Ringcrg 20038  CRingccrg 20039  AssAlgcasa 21378  algSccascl 21380  var₁cv1 21669  Poly₁cpl1 21670  coe₁cco1 21671   Mat cmat 21876   matToPolyMat cmat2pmat 22175   decompPMat cdecpmat 22233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-cnex 11153  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-mulcom 11161  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173  ax-pre-mulgt0 11174

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4905  df-int 4947  df-iun 4995  df-iin 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7657  df-ofr 7658  df-om 7843  df-1st 7962  df-2nd 7963  df-supp 8134  df-frecs 8253  df-wrecs 8284  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8691  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9350  df-sup 9424  df-oi 9492  df-card 9921  df-pnf 11237  df-mnf 11238  df-xr 11239  df-ltxr 11240  df-le 11241  df-sub 11433  df-neg 11434  df-nn 12200  df-2 12262  df-3 12263  df-4 12264  df-5 12265  df-6 12266  df-7 12267  df-8 12268  df-9 12269  df-n0 12460  df-z 12546  df-dec 12665  df-uz 12810  df-fz 13472  df-fzo 13615  df-seq 13954  df-hash 14278  df-struct 17067  df-sets 17084  df-slot 17102  df-ndx 17114  df-base 17132  df-ress 17161  df-plusg 17197  df-mulr 17198  df-sca 17200  df-vsca 17201  df-ip 17202  df-tset 17203  df-ple 17204  df-ds 17206  df-hom 17208  df-cco 17209  df-0g 17374  df-gsum 17375  df-prds 17380  df-pws 17382  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18548  df-sgrp 18597  df-mnd 18613  df-mhm 18658  df-submnd 18659  df-grp 18809  df-minusg 18810  df-sbg 18811  df-mulg 18936  df-subg 18988  df-ghm 19075  df-cntz 19166  df-cmn 19634  df-abl 19635  df-mgp 19971  df-ur 19988  df-ring 20040  df-cring 20041  df-subrg 20338  df-lmod 20450  df-lss 20520  df-sra 20762  df-rgmod 20763  df-dsmm 21260  df-frlm 21275  df-assa 21381  df-ascl 21383  df-psr 21433  df-mvr 21434  df-mpl 21435  df-opsr 21437  df-psr1 21673  df-vr1 21674  df-ply1 21675  df-coe1 21676  df-mamu 21855  df-mat 21877  df-mat2pmat 22178  df-decpmat 22234

This theorem is referenced by:  monmat2matmon  22295