Theorem decpmatid 21380

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 21303	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1128	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5		eqid 2823	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1r‘𝐶)
7	5, 6	ringidcl 19320	. . . 4 ⊢ (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
8	4, 7	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9		simp3 1134	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
10	2, 5	decpmatval 21375	. . 3 ⊢ ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
11	8, 9, 10	syl2anc 586	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12		eqid 2823	. . . . . . 7 ⊢ (0g‘𝑃) = (0g‘𝑃)
13		eqid 2823	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
14		simp11 1199	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
15		simp12 1200	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
16		simp2 1133	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
17		simp3 1134	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
18	1, 2, 12, 13, 14, 15, 16, 17, 6	pmat1ovd 21307	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))
19	18	fveq2d 6676	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))))
20	19	fveq1d 6674	. . . 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 6688	. . . . . . 7 ⊢ (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))
22	21	fveq1i 6673	. . . . . 6 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))‘𝐾)
23		iffv 6689	. . . . . 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 2846	. . . . 5 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾))
25		eqid 2823	. . . . . . . . . . . . 13 ⊢ (var1‘𝑅) = (var1‘𝑅)
26		eqid 2823	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2823	. . . . . . . . . . . . 13 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 20463	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
29	28	3ad2ant2 1130	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) = (1r‘𝑃))
30	29	eqcomd 2829	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
31	30	fveq2d 6676	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r‘𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
32	31	fveq1d 6674	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))‘𝐾))
33	1	ply1lmod 20422	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35		0nn0 11915	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
36		eqid 2823	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 20441	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
38	35, 37	mpan2 689	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
39	38	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
40		eqid 2823	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2823	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42		eqid 2823	. . . . . . . . . . . . 13 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 19664	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 2829	. . . . . . . . . 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 6676	. . . . . . . . 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 6674	. . . . . . . 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 1133	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
49	1	ply1sca 20423	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1130	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2829	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6676	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
53		eqid 2823	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2823	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
55	53, 54	ringidcl 19320	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
56	55	3ad2ant2 1130	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘𝑅) ∈ (Base‘𝑅))
57	52, 56	eqeltrd 2915	. . . . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 20443	. . . . . . . . . 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 1367	. . . . . . . . 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 2827	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4491	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
64	63	adantl 484	. . . . . . . . 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 6685	. . . . . . . . . . 11 ⊢ (1r‘(Scalar‘𝑃)) ∈ V
66		fvex 6685	. . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V
67	65, 66	ifex 4517	. . . . . . . . . 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 6777	. . . . . . . 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 2862	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
71	1, 12, 59	coe1z 20433	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
72	71	3ad2ant2 1130	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g‘𝑃)) = (ℕ0 × {(0g‘𝑅)}))
73	72	fveq1d 6674	. . . . . . . 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 6966	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
76	74, 9, 75	syl2anc 586	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
77	73, 76	eqtrd 2858	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝐾) = (0g‘𝑅))
78	70, 77	ifeq12d 4489	. . . . . 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 1129	. . . . 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	syl5eq 2870	. . . 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 2858	. . 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 7233	. 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 484	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 𝑅 = (Scalar‘𝑃))
84	83	eqcomd 2829	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6676	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
86	85	ifeq1d 4487	. . . . . 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 7235	. . . . 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 4475	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (1r‘(Scalar‘𝑃)))
89	88	ifeq1d 4487	. . . . . . 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 483	. . . . . 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 7235	. . . . 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 21058	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
95	92, 94	syl5eq 2870	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
96	95	3adant3 1128	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
97	96	adantl 484	. . . . 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 2868	. . . 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 4475	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2829	. . . . 5 ⊢ (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101	100	adantr 483	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = if(𝐾 = 0, 1 , 0 ))
102	98, 101	eqtrd 2858	. . 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 4508	. . . . . . 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 7235	. . . . 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 4478	. . . . . . . 8 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (0g‘𝑅))
107	106	adantr 483	. . . . . . 7 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (0g‘𝑅))
108	107	ifeq1d 4487	. . . . . 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 7235	. . . . 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 1144	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
111	110	adantl 484	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
112		decpmatid.0	. . . . . . 7 ⊢ 0 = (0g‘𝐴)
113	93, 59	mat0op 21030	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
114	112, 113	syl5eq 2870	. . . . . 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 2868	. . . 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 4478	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2829	. . . . 5 ⊢ (¬ 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119	118	adantr 483	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = if(𝐾 = 0, 1 , 0 ))
120	116, 119	eqtrd 2858	. . 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 810	. 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 2862	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3496  ifcif 4469  {csn 4569   ↦ cmpt 5148   × cxp 5555  ‘cfv 6357  (class class class)co 7158   ∈ cmpo 7160  Fincfn 8511  0cc0 10539  ℕ₀cn0 11900  Basecbs 16485  Scalarcsca 16570   ·_𝑠 cvsca 16571  0_gc0g 16715  ._gcmg 18226  mulGrpcmgp 19241  1_rcur 19253  Ringcrg 19299  LModclmod 19636  var₁cv1 20346  Poly₁cpl1 20347  coe₁cco1 20348   Mat cmat 21018   decompPMat cdecpmat 21372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-ot 4578  df-uni 4841  df-int 4879  df-iun 4923  df-iin 4924  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-of 7411  df-ofr 7412  df-om 7583  df-1st 7691  df-2nd 7692  df-supp 7833  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-fsupp 8836  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-fz 12896  df-fzo 13037  df-seq 13373  df-hash 13694  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-hom 16591  df-cco 16592  df-0g 16717  df-gsum 16718  df-prds 16723  df-pws 16725  df-mre 16859  df-mrc 16860  df-acs 16862  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-mhm 17958  df-submnd 17959  df-grp 18108  df-minusg 18109  df-sbg 18110  df-mulg 18227  df-subg 18278  df-ghm 18358  df-cntz 18449  df-cmn 18910  df-abl 18911  df-mgp 19242  df-ur 19254  df-ring 19301  df-subrg 19535  df-lmod 19638  df-lss 19706  df-sra 19946  df-rgmod 19947  df-ascl 20089  df-psr 20138  df-mvr 20139  df-mpl 20140  df-opsr 20142  df-psr1 20350  df-vr1 20351  df-ply1 20352  df-coe1 20353  df-dsmm 20878  df-frlm 20893  df-mamu 20997  df-mat 21019  df-decpmat 21373

This theorem is referenced by:  idpm2idmp  21411