Theorem decpmatid 22708

Description: The matrix consisting of the coefficients in the polynomial entries of the identity matrix is an identity or a zero matrix. (Contributed by AV, 28-Sep-2019.) (Revised by AV, 2-Dec-2019.)

Hypotheses

Ref	Expression
decpmatid.p	⊢ 𝑃 = (Poly1‘𝑅)
decpmatid.c	⊢ 𝐶 = (𝑁 Mat 𝑃)
decpmatid.i	⊢ 𝐼 = (1r‘𝐶)
decpmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
decpmatid.0	⊢ 0 = (0g‘𝐴)
decpmatid.1	⊢ 1 = (1r‘𝐴)

Assertion

Ref	Expression
decpmatid	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))

Proof of Theorem decpmatid

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

Step	Hyp	Ref	Expression
1		decpmatid.p	. . . . . 6 ⊢ 𝑃 = (Poly1‘𝑅)
2		decpmatid.c	. . . . . 6 ⊢ 𝐶 = (𝑁 Mat 𝑃)
3	1, 2	pmatring 22630	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5		eqid 2735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1r‘𝐶)
7	5, 6	ringidcl 20225	. . . 4 ⊢ (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
8	4, 7	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9		simp3 1138	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
10	2, 5	decpmatval 22703	. . 3 ⊢ ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
11	8, 9, 10	syl2anc 584	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12		eqid 2735	. . . . . . 7 ⊢ (0g‘𝑃) = (0g‘𝑃)
13		eqid 2735	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
14		simp11 1204	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
15		simp12 1205	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
16		simp2 1137	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
17		simp3 1138	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
18	1, 2, 12, 13, 14, 15, 16, 17, 6	pmat1ovd 22635	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))
19	18	fveq2d 6880	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))))
20	19	fveq1d 6878	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾))
21		fvif 6892	. . . . . . 7 ⊢ (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))
22	21	fveq1i 6877	. . . . . 6 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))‘𝐾)
23		iffv 6893	. . . . . 6 ⊢ (if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾))
24	22, 23	eqtri 2758	. . . . 5 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾))
25		eqid 2735	. . . . . . . . . . . . 13 ⊢ (var1‘𝑅) = (var1‘𝑅)
26		eqid 2735	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2735	. . . . . . . . . . . . 13 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 22232	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
29	28	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
30	29	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
31	30	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r‘𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
32	31	fveq1d 6878	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))‘𝐾))
33	1	ply1lmod 22187	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35		0nn0 12516	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
36		eqid 2735	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 22208	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
38	35, 37	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
39	38	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
40		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2735	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 20847	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LMod ∧ (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
44	34, 39, 43	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
45	44	eqcomd 2741	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = ((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
46	45	fveq2d 6880	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))) = (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))))
47	46	fveq1d 6878	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))‘𝐾) = ((coe1‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))‘𝐾))
48		simp2 1137	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
49	1	ply1sca 22188	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2741	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6880	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
53		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2735	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
55	53, 54	ringidcl 20225	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
56	55	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘𝑅) ∈ (Base‘𝑅))
57	52, 56	eqeltrd 2834	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅))
58	35	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 0 ∈ ℕ0)
59		eqid 2735	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 22210	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅) ∧ 0 ∈ ℕ0) → (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅))))
61	48, 57, 58, 60	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅))))
62		eqeq1 2739	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4524	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
64	63	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 = 𝐾) → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
65		fvex 6889	. . . . . . . . . . 11 ⊢ (1r‘(Scalar‘𝑃)) ∈ V
66		fvex 6889	. . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V
67	65, 66	ifex 4551	. . . . . . . . . 10 ⊢ if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) ∈ V)
69	61, 64, 9, 68	fvmptd 6993	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘((1r‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
70	32, 47, 69	3eqtrd 2774	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
71	1, 12, 59	coe1z 22200	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
72	71	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
73	72	fveq1d 6878	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝐾) = ((ℕ0 × {(0g‘𝑅)})‘𝐾))
74	66	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0g‘𝑅) ∈ V)
75		fvconst2g 7194	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
76	74, 9, 75	syl2anc 584	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
77	73, 76	eqtrd 2770	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝐾) = (0g‘𝑅))
78	70, 77	ifeq12d 4522	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)))
79	78	3ad2ant1 1133	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)))
80	24, 79	eqtrid 2782	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)))
81	20, 80	eqtrd 2770	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)))
82	81	mpoeq3dva 7484	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))))
83	50	adantl 481	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 𝑅 = (Scalar‘𝑃))
84	83	eqcomd 2741	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6880	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
86	85	ifeq1d 4520	. . . . . 6 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅)))
87	86	mpoeq3dv 7486	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
88		iftrue 4506	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (1r‘(Scalar‘𝑃)))
89	88	ifeq1d 4520	. . . . . . 7 ⊢ (𝐾 = 0 → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)) = if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
90	89	adantr 480	. . . . . 6 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)) = if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
91	90	mpoeq3dv 7486	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g‘𝑅))))
92		decpmatid.1	. . . . . . . 8 ⊢ 1 = (1r‘𝐴)
93		decpmatid.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
94	93, 54, 59	mat1 22385	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
95	92, 94	eqtrid 2782	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
96	95	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
97	96	adantl 481	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
98	87, 91, 97	3eqtr4d 2780	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = 1 )
99		iftrue 4506	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2741	. . . . 5 ⊢ (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101	100	adantr 480	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = if(𝐾 = 0, 1 , 0 ))
102	98, 101	eqtrd 2770	. . 3 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
103		ifid 4541	. . . . . . 7 ⊢ if(𝑖 = 𝑗, (0g‘𝑅), (0g‘𝑅)) = (0g‘𝑅)
104	103	a1i 11	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (0g‘𝑅), (0g‘𝑅)) = (0g‘𝑅))
105	104	mpoeq3dv 7486	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (0g‘𝑅), (0g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
106		iffalse 4509	. . . . . . . 8 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (0g‘𝑅))
107	106	adantr 480	. . . . . . 7 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (0g‘𝑅))
108	107	ifeq1d 4520	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅)) = if(𝑖 = 𝑗, (0g‘𝑅), (0g‘𝑅)))
109	108	mpoeq3dv 7486	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (0g‘𝑅), (0g‘𝑅))))
110		3simpa 1148	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
111	110	adantl 481	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
112		decpmatid.0	. . . . . . 7 ⊢ 0 = (0g‘𝐴)
113	93, 59	mat0op 22357	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
114	112, 113	eqtrid 2782	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
115	111, 114	syl 17	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
116	105, 109, 115	3eqtr4d 2780	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = 0 )
117		iffalse 4509	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2741	. . . . 5 ⊢ (¬ 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119	118	adantr 480	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = if(𝐾 = 0, 1 , 0 ))
120	116, 119	eqtrd 2770	. . 3 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
121	102, 120	pm2.61ian 811	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)), (0g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
122	11, 82, 121	3eqtrd 2774	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ifcif 4500  {csn 4601   ↦ cmpt 5201   × cxp 5652  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  Fincfn 8959  0cc0 11129  ℕ₀cn0 12501  Basecbs 17228  Scalarcsca 17274   ·_𝑠 cvsca 17275  0_gc0g 17453  ._gcmg 19050  mulGrpcmgp 20100  1_rcur 20141  Ringcrg 20193  LModclmod 20817  var₁cv1 22111  Poly₁cpl1 22112  coe₁cco1 22113   Mat cmat 22345   decompPMat cdecpmat 22700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7671  df-ofr 7672  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-sup 9454  df-oi 9524  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-fzo 13672  df-seq 14020  df-hash 14349  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-mulr 17285  df-sca 17287  df-vsca 17288  df-ip 17289  df-tset 17290  df-ple 17291  df-ds 17293  df-hom 17295  df-cco 17296  df-0g 17455  df-gsum 17456  df-prds 17461  df-pws 17463  df-mre 17598  df-mrc 17599  df-acs 17601  df-mgm 18618  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-submnd 18762  df-grp 18919  df-minusg 18920  df-sbg 18921  df-mulg 19051  df-subg 19106  df-ghm 19196  df-cntz 19300  df-cmn 19763  df-abl 19764  df-mgp 20101  df-rng 20113  df-ur 20142  df-ring 20195  df-subrng 20506  df-subrg 20530  df-lmod 20819  df-lss 20889  df-sra 21131  df-rgmod 21132  df-dsmm 21692  df-frlm 21707  df-psr 21869  df-mvr 21870  df-mpl 21871  df-opsr 21873  df-psr1 22115  df-vr1 22116  df-ply1 22117  df-coe1 22118  df-mamu 22329  df-mat 22346  df-decpmat 22701

This theorem is referenced by:  idpm2idmp  22739