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Theorem decpmatid 22263

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 22185	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
4	3	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝐶 ∈ Ring)
5		eqid 2732	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
6		decpmatid.i	. . . . 5 ⊢ 𝐼 = (1_r‘𝐶)
7	5, 6	ringidcl 20076	. . . 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 22258	. . 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 2732	. . . . . . 7 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
13		eqid 2732	. . . . . . 7 ⊢ (1_r‘𝑃) = (1_r‘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 22190	. . . . . 6 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1_r‘𝑃), (0_g‘𝑃)))
19	18	fveq2d 6892	. . . . 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 6890	. . . 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 6904	. . . . . . 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 6889	. . . . . 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 6905	. . . . . 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 2760	. . . . 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 2732	. . . . . . . . . . . . 13 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
26		eqid 2732	. . . . . . . . . . . . 13 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
27		eqid 2732	. . . . . . . . . . . . 13 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
28	1, 25, 26, 27	ply1idvr1 21808	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (1_r‘𝑃))
29	28	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (1_r‘𝑃))
30	29	eqcomd 2738	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘𝑃) = (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
31	30	fveq2d 6892	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(1_r‘𝑃)) = (coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
32	31	fveq1d 6890	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(1_r‘𝑃))‘𝐾) = ((coe₁‘(0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))‘𝐾))
33	1	ply1lmod 21765	. . . . . . . . . . . . 13 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34	33	3ad2ant2 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑃 ∈ LMod)
35		0nn0 12483	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℕ₀
36		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
37	1, 25, 26, 27, 36	ply1moncl 21784	. . . . . . . . . . . . . 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 1134	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (0(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
40		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
41		eqid 2732	. . . . . . . . . . . . 13 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
42		eqid 2732	. . . . . . . . . . . . 13 ⊢ (1_r‘(Scalar‘𝑃)) = (1_r‘(Scalar‘𝑃))
43	36, 40, 41, 42	lmodvs1 20492	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 2738	. . . . . . . . . 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 6892	. . . . . . . . 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 6890	. . . . . . . 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 1137	. . . . . . . . . 10 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑅 ∈ Ring)
49	1	ply1sca 21766	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50	49	3ad2ant2 1134	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 𝑅 = (Scalar‘𝑃))
51	50	eqcomd 2738	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (Scalar‘𝑃) = 𝑅)
52	51	fveq2d 6892	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
53		eqid 2732	. . . . . . . . . . . . 13 ⊢ (Base‘𝑅) = (Base‘𝑅)
54		eqid 2732	. . . . . . . . . . . . 13 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
55	53, 54	ringidcl 20076	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ Ring → (1_r‘𝑅) ∈ (Base‘𝑅))
56	55	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (1_r‘𝑅) ∈ (Base‘𝑅))
57	52, 56	eqeltrd 2833	. . . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
60	59, 53, 1, 25, 41, 26, 27	coe1tm 21786	. . . . . . . . . 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 1371	. . . . . . . . 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 2736	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
63	62	ifbid 4550	. . . . . . . . . 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 482	. . . . . . . . 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 6901	. . . . . . . . . . 11 ⊢ (1_r‘(Scalar‘𝑃)) ∈ V
66		fvex 6901	. . . . . . . . . . 11 ⊢ (0_g‘𝑅) ∈ V
67	65, 66	ifex 4577	. . . . . . . . . 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 7002	. . . . . . . 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 2776	. . . . . . 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 21776	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → (coe₁‘(0_g‘𝑃)) = (ℕ₀ × {(0_g‘𝑅)}))
72	71	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (coe₁‘(0_g‘𝑃)) = (ℕ₀ × {(0_g‘𝑅)}))
73	72	fveq1d 6890	. . . . . . . 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 584	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((ℕ₀ × {(0_g‘𝑅)})‘𝐾) = (0_g‘𝑅))
77	73, 76	eqtrd 2772	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → ((coe₁‘(0_g‘𝑃))‘𝐾) = (0_g‘𝑅))
78	70, 77	ifeq12d 4548	. . . . . 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 1133	. . . . 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 2784	. . . 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 2772	. . 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 482	. . . . . . . . 9 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 𝑅 = (Scalar‘𝑃))
84	83	eqcomd 2738	. . . . . . . 8 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (Scalar‘𝑃) = 𝑅)
85	84	fveq2d 6892	. . . . . . 7 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → (1_r‘(Scalar‘𝑃)) = (1_r‘𝑅))
86	85	ifeq1d 4546	. . . . . 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 4533	. . . . . . . 8 ⊢ (𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (1_r‘(Scalar‘𝑃)))
89	88	ifeq1d 4546	. . . . . . 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 481	. . . . . 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 21940	. . . . . . . 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 2784	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
96	95	3adant3 1132	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → 1 = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝑗, (1_r‘𝑅), (0_g‘𝑅))))
97	96	adantl 482	. . . . 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 2782	. . . 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 4533	. . . . . 6 ⊢ (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
100	99	eqcomd 2738	. . . . 5 ⊢ (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101	100	adantr 481	. . . 4 ⊢ ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 1 = if(𝐾 = 0, 1 , 0 ))
102	98, 101	eqtrd 2772	. . 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 4567	. . . . . . 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 4536	. . . . . . . 8 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, (1_r‘(Scalar‘𝑃)), (0_g‘𝑅)) = (0_g‘𝑅))
107	106	adantr 481	. . . . . . 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 4546	. . . . . 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 1148	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
111	110	adantl 482	. . . . . 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 21912	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0_g‘𝐴) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (0_g‘𝑅)))
114	112, 113	eqtrid 2784	. . . . . 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 2782	. . . 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 4536	. . . . . 6 ⊢ (¬ 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118	117	eqcomd 2738	. . . . 5 ⊢ (¬ 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119	118	adantr 481	. . . 4 ⊢ ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ₀)) → 0 = if(𝐾 = 0, 1 , 0 ))
120	116, 119	eqtrd 2772	. . 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 2776	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ifcif 4527 {csn 4627 ↦ cmpt 5230 × cxp 5673 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 Fincfn 8935 0cc0 11106 ℕ₀cn0 12468 Basecbs 17140 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 ._gcmg 18944 mulGrpcmgp 19981 1_rcur 19998 Ringcrg 20049 LModclmod 20463 var₁cv1 21691 Poly₁cpl1 21692 coe₁cco1 21693 Mat cmat 21898 decompPMat cdecpmat 22255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-subrg 20353 df-lmod 20465 df-lss 20535 df-sra 20777 df-rgmod 20778 df-dsmm 21278 df-frlm 21293 df-ascl 21401 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mamu 21877 df-mat 21899 df-decpmat 22256

This theorem is referenced by: idpm2idmp 22294

W3C validator