Theorem decpmatid 22797

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 22719	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5		eqid 2740	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1r‘𝐶)
7	5, 6	ringidcl 20289	. . . 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 22792	. . 3 ⊢ ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
11	8, 9, 10	syl2anc 583	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12		eqid 2740	. . . . . . 7 ⊢ (0g‘𝑃) = (0g‘𝑃)
13		eqid 2740	. . . . . . 7 ⊢ (1r‘𝑃) = (1r‘𝑃)
14		simp11 1203	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
15		simp12 1204	. . . . . . 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 22724	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))
19	18	fveq2d 6924	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))))
20	19	fveq1d 6922	. . . 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 6936	. . . . . . 7 ⊢ (coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))
22	21	fveq1i 6921	. . . . . 6 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r‘𝑃)), (coe1‘(0g‘𝑃)))‘𝐾)
23		iffv 6937	. . . . . 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 2768	. . . . 5 ⊢ ((coe1‘if(𝑖 = 𝑗, (1r‘𝑃), (0g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r‘𝑃))‘𝐾), ((coe1‘(0g‘𝑃))‘𝐾))
25		eqid 2740	. . . . . . . . . . . . 13 ⊢ (var1‘𝑅) = (var1‘𝑅)
26		eqid 2740	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2740	. . . . . . . . . . . . 13 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 22319	. . . . . . . . . . . 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 2746	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘𝑃) = (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))
31	30	fveq2d 6924	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r‘𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅))))
32	31	fveq1d 6922	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1‘𝑅)))‘𝐾))
33	1	ply1lmod 22274	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35		0nn0 12568	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ0
36		eqid 2740	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 22295	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1‘𝑅)) ∈ (Base‘𝑃))
38	35, 37	mpan2 690	. . . . . . . . . . . . 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 2740	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2740	. . . . . . . . . . . . 13 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
42		eqid 2740	. . . . . . . . . . . . 13 ⊢ (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 20910	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 2746	. . . . . . . . . 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 6924	. . . . . . . . 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 6922	. . . . . . . 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 22275	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2746	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6924	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
53		eqid 2740	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2740	. . . . . . . . . . . . 13 ⊢ (1r‘𝑅) = (1r‘𝑅)
55	53, 54	ringidcl 20289	. . . . . . . . . . . 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 2844	. . . . . . . . . 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 2740	. . . . . . . . . . 11 ⊢ (0g‘𝑅) = (0g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 22297	. . . . . . . . . 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 1371	. . . . . . . . 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 2744	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4571	. . . . . . . . . 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 6933	. . . . . . . . . . 11 ⊢ (1r‘(Scalar‘𝑃)) ∈ V
66		fvex 6933	. . . . . . . . . . 11 ⊢ (0g‘𝑅) ∈ V
67	65, 66	ifex 4598	. . . . . . . . . 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 7036	. . . . . . . 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 2784	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r‘𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)))
71	1, 12, 59	coe1z 22287	. . . . . . . . . 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 6922	. . . . . . . 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 7239	. . . . . . . . 9 ⊢ (((0g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
76	74, 9, 75	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g‘𝑅)})‘𝐾) = (0g‘𝑅))
77	73, 76	eqtrd 2780	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g‘𝑃))‘𝐾) = (0g‘𝑅))
78	70, 77	ifeq12d 4569	. . . . . 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 2792	. . . 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 2780	. . 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 7527	. 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 2746	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6924	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r‘𝑅))
86	85	ifeq1d 4567	. . . . . 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 7529	. . . . 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 4554	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g‘𝑅)) = (1r‘(Scalar‘𝑃)))
89	88	ifeq1d 4567	. . . . . . 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 7529	. . . . 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 22474	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1r‘𝑅), (0g‘𝑅))))
95	92, 94	eqtrid 2792	. . . . . . 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 2790	. . . 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 4554	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2746	. . . . 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 2780	. . 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 4588	. . . . . . 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 7529	. . . . 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 4557	. . . . . . . 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 4567	. . . . . 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 7529	. . . . 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 22446	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0g‘𝑅)))
114	112, 113	eqtrid 2792	. . . . . 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 2790	. . . 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 4557	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2746	. . . . 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 2780	. . 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 2784	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ifcif 4548  {csn 4648   ↦ cmpt 5249   × cxp 5698  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  Fincfn 9003  0cc0 11184  ℕ₀cn0 12553  Basecbs 17258  Scalarcsca 17314   ·_𝑠 cvsca 17315  0_gc0g 17499  ._gcmg 19107  mulGrpcmgp 20161  1_rcur 20208  Ringcrg 20260  LModclmod 20880  var₁cv1 22198  Poly₁cpl1 22199  coe₁cco1 22200   Mat cmat 22432   decompPMat cdecpmat 22789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714  df-ofr 7715  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-er 8763  df-map 8886  df-pm 8887  df-ixp 8956  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-fsupp 9432  df-sup 9511  df-oi 9579  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-4 12358  df-5 12359  df-6 12360  df-7 12361  df-8 12362  df-9 12363  df-n0 12554  df-z 12640  df-dec 12759  df-uz 12904  df-fz 13568  df-fzo 13712  df-seq 14053  df-hash 14380  df-struct 17194  df-sets 17211  df-slot 17229  df-ndx 17241  df-base 17259  df-ress 17288  df-plusg 17324  df-mulr 17325  df-sca 17327  df-vsca 17328  df-ip 17329  df-tset 17330  df-ple 17331  df-ds 17333  df-hom 17335  df-cco 17336  df-0g 17501  df-gsum 17502  df-prds 17507  df-pws 17509  df-mre 17644  df-mrc 17645  df-acs 17647  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-mhm 18818  df-submnd 18819  df-grp 18976  df-minusg 18977  df-sbg 18978  df-mulg 19108  df-subg 19163  df-ghm 19253  df-cntz 19357  df-cmn 19824  df-abl 19825  df-mgp 20162  df-rng 20180  df-ur 20209  df-ring 20262  df-subrng 20572  df-subrg 20597  df-lmod 20882  df-lss 20953  df-sra 21195  df-rgmod 21196  df-dsmm 21775  df-frlm 21790  df-psr 21952  df-mvr 21953  df-mpl 21954  df-opsr 21956  df-psr1 22202  df-vr1 22203  df-ply1 22204  df-coe1 22205  df-mamu 22416  df-mat 22433  df-decpmat 22790

This theorem is referenced by:  idpm2idmp  22828