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Theorem decpmatid 22627

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 = (1_r‘𝐶)
decpmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
decpmatid.0	⊢ 0 = (0_g‘𝐴)
decpmatid.1	⊢ 1 = (1_r‘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 22549	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1129	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝐶 ∈ Ring)
5		eqid 2726	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1_r‘𝐶)
7	5, 6	ringidcl 20165	. . . 4 ⊢ (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
8	4, 7	syl 17	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝐼 ∈ (Base‘𝐶))
9		simp3 1135	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝐾 ∈ ℕ₀)
10	2, 5	decpmatval 22622	. . 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 2726	. . . . . . 7 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
13		eqid 2726	. . . . . . 7 ⊢ (1_r‘𝑃) = (1_r‘𝑃)
14		simp11 1200	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑁 ∈ Fin)
15		simp12 1201	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
16		simp2 1134	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
17		simp3 1135	. . . . . . 7 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
18	1, 2, 12, 13, 14, 15, 16, 17, 6	pmat1ovd 22554	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))
19	18	fveq2d 6889	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (coe₁‘(𝑖𝐼𝑗)) = (coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃))))
20	19	fveq1d 6887	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe₁‘(𝑖𝐼𝑗))‘𝐾) = ((coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))‘𝐾))
21		fvif 6901	. . . . . . 7 ⊢ (coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃))) = if(𝑖 = 𝑗, (coe₁‘(1_r‘𝑃)), (coe₁‘(0_g‘𝑃)))
22	21	fveq1i 6886	. . . . . 6 ⊢ ((coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe₁‘(1_r‘𝑃)), (coe₁‘(0_g‘𝑃)))‘𝐾)
23		iffv 6902	. . . . . 6 ⊢ (if(𝑖 = 𝑗, (coe₁‘(1_r‘𝑃)), (coe₁‘(0_g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe₁‘(1_r‘𝑃))‘𝐾), ((coe₁‘(0_g‘𝑃))‘𝐾))
24	22, 23	eqtri 2754	. . . . 5 ⊢ ((coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe₁‘(1_r‘𝑃))‘𝐾), ((coe₁‘(0_g‘𝑃))‘𝐾))
25		eqid 2726	. . . . . . . . . . . . 13 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
26		eqid 2726	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2726	. . . . . . . . . . . . 13 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 22169	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (1_r‘𝑃))
29	28	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (1_r‘𝑃))
30	29	eqcomd 2732	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘𝑃) = (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
31	30	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(1_r‘𝑃)) = (coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
32	31	fveq1d 6887	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(1_r‘𝑃))‘𝐾) = ((coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))‘𝐾))
33	1	ply1lmod 22125	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑃 ∈ LMod)
35		0nn0 12491	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
36		eqid 2726	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 22145	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ Ring ∧ 0 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
38	35, 37	mpan2 688	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
39	38	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
40		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2726	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
42		eqid 2726	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑃)) = (1_r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 20736	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LMod ∧ (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃)) → ((1_r‘(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) → ((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
45	44	eqcomd 2732	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = ((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
46	45	fveq2d 6889	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (coe₁‘((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))
47	46	fveq1d 6887	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))‘𝐾) = ((coe₁‘((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝐾))
48		simp2 1134	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑅 ∈ Ring)
49	1	ply1sca 22126	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2732	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6889	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
53		eqid 2726	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2726	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
55	53, 54	ringidcl 20165	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
56	55	3ad2ant2 1131	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘𝑅) ∈ (Base‘𝑅))
57	52, 56	eqeltrd 2827	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘(Scalar‘𝑃)) ∈ (Base‘𝑅))
58	35	a1i 11	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 0 ∈ ℕ₀)
59		eqid 2726	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 22147	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ (1_r‘(Scalar‘𝑃)) ∈ (Base‘𝑅) ∧ 0 ∈ ℕ₀) → (coe₁‘((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) = (𝑘 ∈ ℕ₀ ↦ if(𝑘 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅))))
61	48, 57, 58, 60	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) = (𝑘 ∈ ℕ₀ ↦ if(𝑘 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅))))
62		eqeq1 2730	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4546	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → if(𝑘 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
64	63	adantl 481	. . . . . . . . 9 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑘 = 𝐾) → if(𝑘 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
65		fvex 6898	. . . . . . . . . . 11 ⊢ (1_r‘(Scalar‘𝑃)) ∈ V
66		fvex 6898	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) ∈ V
67	65, 66	ifex 4573	. . . . . . . . . 10 ⊢ if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) ∈ V
68	67	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) ∈ V)
69	61, 64, 9, 68	fvmptd 6999	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘((1_r‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝐾) = if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
70	32, 47, 69	3eqtrd 2770	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(1_r‘𝑃))‘𝐾) = if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
71	1, 12, 59	coe1z 22137	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe₁‘(0_g‘𝑃)) = (ℕ₀ × {(0_g‘𝑅)}))
72	71	3ad2ant2 1131	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(0_g‘𝑃)) = (ℕ₀ × {(0_g‘𝑅)}))
73	72	fveq1d 6887	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(0_g‘𝑃))‘𝐾) = ((ℕ₀ × {(0_g‘𝑅)})‘𝐾))
74	66	a1i 11	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0_g‘𝑅) ∈ V)
75		fvconst2g 7199	. . . . . . . . 9 ⊢ (((0_g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ₀) → ((ℕ₀ × {(0_g‘𝑅)})‘𝐾) = (0_g‘𝑅))
76	74, 9, 75	syl2anc 583	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((ℕ₀ × {(0_g‘𝑅)})‘𝐾) = (0_g‘𝑅))
77	73, 76	eqtrd 2766	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(0_g‘𝑃))‘𝐾) = (0_g‘𝑅))
78	70, 77	ifeq12d 4544	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → if(𝑖 = 𝑗, ((coe₁‘(1_r‘𝑃))‘𝐾), ((coe₁‘(0_g‘𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)))
79	78	3ad2ant1 1130	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝑗, ((coe₁‘(1_r‘𝑃))‘𝐾), ((coe₁‘(0_g‘𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)))
80	24, 79	eqtrid 2778	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe₁‘if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)))
81	20, 80	eqtrd 2766	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → ((coe₁‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)))
82	81	mpoeq3dva 7482	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe₁‘(𝑖𝐼𝑗))‘𝐾)) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))))
83	50	adantl 481	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 𝑅 = (Scalar‘𝑃))
84	83	eqcomd 2732	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6889	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
86	85	ifeq1d 4542	. . . . . 6 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → if(𝑖 = 𝑗, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅)))
87	86	mpoeq3dv 7484	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
88		iftrue 4529	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (1_r‘(Scalar‘𝑃)))
89	88	ifeq1d 4542	. . . . . . 7 ⊢ (𝐾 = 0 → if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)) = if(𝑖 = 𝑗, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
90	89	adantr 480	. . . . . 6 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)) = if(𝑖 = 𝑗, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)))
91	90	mpoeq3dv 7484	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅))))
92		decpmatid.1	. . . . . . . 8 ⊢ 1 = (1_r‘𝐴)
93		decpmatid.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
94	93, 54, 59	mat1 22304	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1_r‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
95	92, 94	eqtrid 2778	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
96	95	3adant3 1129	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
97	96	adantl 481	. . . . 5 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
98	87, 91, 97	3eqtr4d 2776	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = 1 )
99		iftrue 4529	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2732	. . . . 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 2766	. . 3 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
103		ifid 4563	. . . . . . 7 ⊢ if(𝑖 = 𝑗, (0_g‘𝑅), (0_g‘𝑅)) = (0_g‘𝑅)
104	103	a1i 11	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → if(𝑖 = 𝑗, (0_g‘𝑅), (0_g‘𝑅)) = (0_g‘𝑅))
105	104	mpoeq3dv 7484	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (0_g‘𝑅), (0_g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
106		iffalse 4532	. . . . . . . 8 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (0_g‘𝑅))
107	106	adantr 480	. . . . . . 7 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (0_g‘𝑅))
108	107	ifeq1d 4542	. . . . . 6 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅)) = if(𝑖 = 𝑗, (0_g‘𝑅), (0_g‘𝑅)))
109	108	mpoeq3dv 7484	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (0_g‘𝑅), (0_g‘𝑅))))
110		3simpa 1145	. . . . . . 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 = (0_g‘𝐴)
113	93, 59	mat0op 22276	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
114	112, 113	eqtrid 2778	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
115	111, 114	syl 17	. . . . 5 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 0 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
116	105, 109, 115	3eqtr4d 2776	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = 0 )
117		iffalse 4532	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2732	. . . . 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 2766	. . 3 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
121	102, 120	pm2.61ian 809	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)), (0_g‘𝑅))) = if(𝐾 = 0, 1 , 0 ))
122	11, 82, 121	3eqtrd 2770	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (𝐼 decompPMat 𝐾) = if(𝐾 = 0, 1 , 0 ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ifcif 4523 {csn 4623 ↦ cmpt 5224 × cxp 5667 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8941 0cc0 11112 ℕ₀cn0 12476 Basecbs 17153 Scalarcsca 17209 ·_𝑠 cvsca 17210 0_gc0g 17394 ._gcmg 18995 mulGrpcmgp 20039 1_rcur 20086 Ringcrg 20138 LModclmod 20706 var₁cv1 22050 Poly₁cpl1 22051 coe₁cco1 22052 Mat cmat 22262 decompPMat cdecpmat 22619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-ot 4632 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-ofr 7668 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-hom 17230 df-cco 17231 df-0g 17396 df-gsum 17397 df-prds 17402 df-pws 17404 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-sra 21021 df-rgmod 21022 df-dsmm 21627 df-frlm 21642 df-ascl 21750 df-psr 21803 df-mvr 21804 df-mpl 21805 df-opsr 21807 df-psr1 22054 df-vr1 22055 df-ply1 22056 df-coe1 22057 df-mamu 22241 df-mat 22263 df-decpmat 22620

This theorem is referenced by: idpm2idmp 22658

W3C validator