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Theorem decpmatid 22690

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 22612	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1129	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝐶 ∈ Ring)
5		eqid 2725	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1_r‘𝐶)
7	5, 6	ringidcl 20206	. . . 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 22685	. . 3 ⊢ ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ₀) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe₁‘(𝑖𝐼𝑗))‘𝐾)))
11	8, 9, 10	syl2anc 582	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (𝐼 decompPMat 𝐾) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ ((coe₁‘(𝑖𝐼𝑗))‘𝐾)))
12		eqid 2725	. . . . . . 7 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
13		eqid 2725	. . . . . . 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 22617	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))
19	18	fveq2d 6896	. . . . 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 6894	. . . 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 6908	. . . . . . 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 6893	. . . . . 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 6909	. . . . . 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 2753	. . . . 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 2725	. . . . . . . . . . . . 13 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
26		eqid 2725	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2725	. . . . . . . . . . . . 13 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 22223	. . . . . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘𝑃) = (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
31	30	fveq2d 6896	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(1_r‘𝑃)) = (coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
32	31	fveq1d 6894	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(1_r‘𝑃))‘𝐾) = ((coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))‘𝐾))
33	1	ply1lmod 22179	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑃 ∈ LMod)
35		0nn0 12517	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
36		eqid 2725	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 22199	. . . . . . . . . . . . . 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 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
40		eqid 2725	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2725	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
42		eqid 2725	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑃)) = (1_r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 20777	. . . . . . . . . . . 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 582	. . . . . . . . . . 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 2731	. . . . . . . . . 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 6896	. . . . . . . . 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 6894	. . . . . . . 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 22180	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1131	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2731	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6896	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
53		eqid 2725	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2725	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
55	53, 54	ringidcl 20206	. . . . . . . . . . . 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 2825	. . . . . . . . . 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 2725	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 22201	. . . . . . . . . 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 2729	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4547	. . . . . . . . . 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 480	. . . . . . . . 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 6905	. . . . . . . . . . 11 ⊢ (1_r‘(Scalar‘𝑃)) ∈ V
66		fvex 6905	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) ∈ V
67	65, 66	ifex 4574	. . . . . . . . . 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 7007	. . . . . . . 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 2769	. . . . . . 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 22191	. . . . . . . . . 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 6894	. . . . . . . 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 7210	. . . . . . . . 9 ⊢ (((0_g‘𝑅) ∈ V ∧ 𝐾 ∈ ℕ₀) → ((ℕ₀ × {(0_g‘𝑅)})‘𝐾) = (0_g‘𝑅))
76	74, 9, 75	syl2anc 582	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((ℕ₀ × {(0_g‘𝑅)})‘𝐾) = (0_g‘𝑅))
77	73, 76	eqtrd 2765	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(0_g‘𝑃))‘𝐾) = (0_g‘𝑅))
78	70, 77	ifeq12d 4545	. . . . . 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 2777	. . . 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 2765	. . 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 7494	. 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 480	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 𝑅 = (Scalar‘𝑃))
84	83	eqcomd 2731	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6896	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
86	85	ifeq1d 4543	. . . . . 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 7496	. . . . 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 4530	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (1_r‘(Scalar‘𝑃)))
89	88	ifeq1d 4543	. . . . . . 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 479	. . . . . 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 7496	. . . . 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 22367	. . . . . . . 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 2777	. . . . . . 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 480	. . . . 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 2775	. . . 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 4530	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2731	. . . . 5 ⊢ (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101	100	adantr 479	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 1 = if(𝐾 = 0, 1 , 0 ))
102	98, 101	eqtrd 2765	. . 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 4564	. . . . . . 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 7496	. . . . 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 4533	. . . . . . . 8 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (0_g‘𝑅))
107	106	adantr 479	. . . . . . 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 4543	. . . . . 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 7496	. . . . 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 480	. . . . . 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 22339	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
114	112, 113	eqtrid 2777	. . . . . 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 2775	. . . 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 4533	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2731	. . . . 5 ⊢ (¬ 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119	118	adantr 479	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 0 = if(𝐾 = 0, 1 , 0 ))
120	116, 119	eqtrd 2765	. . 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 810	. 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 2769	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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ifcif 4524 {csn 4624 ↦ cmpt 5226 × cxp 5670 ‘cfv 6543 (class class class)co 7416 ∈ cmpo 7418 Fincfn 8962 0cc0 11138 ℕ₀cn0 12502 Basecbs 17179 Scalarcsca 17235 ·_𝑠 cvsca 17236 0_gc0g 17420 ._gcmg 19027 mulGrpcmgp 20078 1_rcur 20125 Ringcrg 20177 LModclmod 20747 var₁cv1 22103 Poly₁cpl1 22104 coe₁cco1 22105 Mat cmat 22325 decompPMat cdecpmat 22682

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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-ofr 7683 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-sup 9465 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-gsum 17423 df-prds 17428 df-pws 17430 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-ghm 19172 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-subrng 20487 df-subrg 20512 df-lmod 20749 df-lss 20820 df-sra 21062 df-rgmod 21063 df-dsmm 21670 df-frlm 21685 df-ascl 21793 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22107 df-vr1 22108 df-ply1 22109 df-coe1 22110 df-mamu 22309 df-mat 22326 df-decpmat 22683

This theorem is referenced by: idpm2idmp 22721

W3C validator