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Theorem decpmatid 22119
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 = (Poly1𝑅)
decpmatid.c 𝐶 = (𝑁 Mat 𝑃)
decpmatid.i 𝐼 = (1r𝐶)
decpmatid.a 𝐴 = (𝑁 Mat 𝑅)
decpmatid.0 0 = (0g𝐴)
decpmatid.1 1 = (1r𝐴)
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.
StepHypRef Expression
1 decpmatid.p . . . . . 6 𝑃 = (Poly1𝑅)
2 decpmatid.c . . . . . 6 𝐶 = (𝑁 Mat 𝑃)
31, 2pmatring 22041 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
433adant3 1132 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5 eqid 2736 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 decpmatid.i . . . . 5 𝐼 = (1r𝐶)
75, 6ringidcl 19989 . . . 4 (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
84, 7syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9 simp3 1138 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
102, 5decpmatval 22114 . . 3 ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
118, 9, 10syl2anc 584 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12 eqid 2736 . . . . . . 7 (0g𝑃) = (0g𝑃)
13 eqid 2736 . . . . . . 7 (1r𝑃) = (1r𝑃)
14 simp11 1203 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
15 simp12 1204 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
16 simp2 1137 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
17 simp3 1138 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 22046 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))
1918fveq2d 6846 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))))
2019fveq1d 6844 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾))
21 fvif 6858 . . . . . . 7 (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))
2221fveq1i 6843 . . . . . 6 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾)
23 iffv 6859 . . . . . 6 (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
2422, 23eqtri 2764 . . . . 5 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
25 eqid 2736 . . . . . . . . . . . . 13 (var1𝑅) = (var1𝑅)
26 eqid 2736 . . . . . . . . . . . . 13 (mulGrp‘𝑃) = (mulGrp‘𝑃)
27 eqid 2736 . . . . . . . . . . . . 13 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
281, 25, 26, 27ply1idvr1 21664 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
29283ad2ant2 1134 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
3029eqcomd 2742 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
3130fveq2d 6846 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
3231fveq1d 6844 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅)))‘𝐾))
331ply1lmod 21623 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34333ad2ant2 1134 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35 0nn0 12428 . . . . . . . . . . . . . 14 0 ∈ ℕ0
36 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
371, 25, 26, 27, 36ply1moncl 21642 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
3835, 37mpan2 689 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
39383ad2ant2 1134 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
40 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2736 . . . . . . . . . . . . 13 ( ·𝑠𝑃) = ( ·𝑠𝑃)
42 eqid 2736 . . . . . . . . . . . . 13 (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
4336, 40, 41, 42lmodvs1 20350 . . . . . . . . . . . 12 ((𝑃 ∈ LMod ∧ (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃)) → ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
4434, 39, 43syl2anc 584 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
4544eqcomd 2742 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
4645fveq2d 6846 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
4746fveq1d 6844 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅)))‘𝐾) = ((coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝐾))
48 simp2 1137 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
491ply1sca 21624 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50493ad2ant2 1134 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
5150eqcomd 2742 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
5251fveq2d 6846 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
53 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2736 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
5553, 54ringidcl 19989 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
56553ad2ant2 1134 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑅) ∈ (Base‘𝑅))
5752, 56eqeltrd 2838 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅))
5835a1i 11 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 0 ∈ ℕ0)
59 eqid 2736 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
6059, 53, 1, 25, 41, 26, 27coe1tm 21644 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (1r‘(Scalar‘𝑃)) ∈ (Base‘𝑅) ∧ 0 ∈ ℕ0) → (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅))))
6148, 57, 58, 60syl3anc 1371 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑘 ∈ ℕ0 ↦ if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅))))
62 eqeq1 2740 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
6362ifbid 4509 . . . . . . . . . 10 (𝑘 = 𝐾 → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
6463adantl 482 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑘 = 𝐾) → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
65 fvex 6855 . . . . . . . . . . 11 (1r‘(Scalar‘𝑃)) ∈ V
66 fvex 6855 . . . . . . . . . . 11 (0g𝑅) ∈ V
6765, 66ifex 4536 . . . . . . . . . 10 if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) ∈ V
6867a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) ∈ V)
6961, 64, 9, 68fvmptd 6955 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
7032, 47, 693eqtrd 2780 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
711, 12, 59coe1z 21634 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
72713ad2ant2 1134 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
7372fveq1d 6844 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = ((ℕ0 × {(0g𝑅)})‘𝐾))
7466a1i 11 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0g𝑅) ∈ V)
75 fvconst2g 7151 . . . . . . . . 9 (((0g𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7674, 9, 75syl2anc 584 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7773, 76eqtrd 2776 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = (0g𝑅))
7870, 77ifeq12d 4507 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
79783ad2ant1 1133 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾)) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8024, 79eqtrid 2788 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8120, 80eqtrd 2776 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8281mpoeq3dva 7434 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))))
8350adantl 482 . . . . . . . . 9 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 𝑅 = (Scalar‘𝑃))
8483eqcomd 2742 . . . . . . . 8 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
8584fveq2d 6846 . . . . . . 7 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
8685ifeq1d 4505 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
8786mpoeq3dv 7436 . . . . 5 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
88 iftrue 4492 . . . . . . . 8 (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (1r‘(Scalar‘𝑃)))
8988ifeq1d 4505 . . . . . . 7 (𝐾 = 0 → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)) = if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)))
9089adantr 481 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)) = if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)))
9190mpoeq3dv 7436 . . . . 5 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅))))
92 decpmatid.1 . . . . . . . 8 1 = (1r𝐴)
93 decpmatid.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
9493, 54, 59mat1 21796 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9592, 94eqtrid 2788 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
96953adant3 1132 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9796adantl 482 . . . . 5 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9887, 91, 973eqtr4d 2786 . . . 4 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = 1 )
99 iftrue 4492 . . . . . 6 (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
10099eqcomd 2742 . . . . 5 (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101100adantr 481 . . . 4 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = if(𝐾 = 0, 1 , 0 ))
10298, 101eqtrd 2776 . . 3 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = if(𝐾 = 0, 1 , 0 ))
103 ifid 4526 . . . . . . 7 if(𝑖 = 𝑗, (0g𝑅), (0g𝑅)) = (0g𝑅)
104103a1i 11 . . . . . 6 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (0g𝑅), (0g𝑅)) = (0g𝑅))
105104mpoeq3dv 7436 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
106 iffalse 4495 . . . . . . . 8 𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (0g𝑅))
107106adantr 481 . . . . . . 7 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (0g𝑅))
108107ifeq1d 4505 . . . . . 6 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)) = if(𝑖 = 𝑗, (0g𝑅), (0g𝑅)))
109108mpoeq3dv 7436 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))))
110 3simpa 1148 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
111110adantl 482 . . . . . 6 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
112 decpmatid.0 . . . . . . 7 0 = (0g𝐴)
11393, 59mat0op 21768 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
114112, 113eqtrid 2788 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 0 = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
115111, 114syl 17 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
116105, 109, 1153eqtr4d 2786 . . . 4 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = 0 )
117 iffalse 4495 . . . . . 6 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118117eqcomd 2742 . . . . 5 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119118adantr 481 . . . 4 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = if(𝐾 = 0, 1 , 0 ))
120116, 119eqtrd 2776 . . 3 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = if(𝐾 = 0, 1 , 0 ))
121102, 120pm2.61ian 810 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅))) = if(𝐾 = 0, 1 , 0 ))
12211, 82, 1213eqtrd 2780 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 3445  ifcif 4486  {csn 4586  cmpt 5188   × cxp 5631  cfv 6496  (class class class)co 7357  cmpo 7359  Fincfn 8883  0cc0 11051  0cn0 12413  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321  .gcmg 18872  mulGrpcmgp 19896  1rcur 19913  Ringcrg 19964  LModclmod 20322  var1cv1 21547  Poly1cpl1 21548  coe1cco1 21549   Mat cmat 21754   decompPMat cdecpmat 22111
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-ot 4595  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-subrg 20220  df-lmod 20324  df-lss 20393  df-sra 20633  df-rgmod 20634  df-dsmm 21138  df-frlm 21153  df-ascl 21261  df-psr 21311  df-mvr 21312  df-mpl 21313  df-opsr 21315  df-psr1 21551  df-vr1 21552  df-ply1 21553  df-coe1 21554  df-mamu 21733  df-mat 21755  df-decpmat 22112
This theorem is referenced by:  idpm2idmp  22150
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