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Theorem decpmatid 22791
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 22713 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
433adant3 1131 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5 eqid 2734 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 decpmatid.i . . . . 5 𝐼 = (1r𝐶)
75, 6ringidcl 20279 . . . 4 (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
84, 7syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9 simp3 1137 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
102, 5decpmatval 22786 . . 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 2734 . . . . . . 7 (0g𝑃) = (0g𝑃)
13 eqid 2734 . . . . . . 7 (1r𝑃) = (1r𝑃)
14 simp11 1202 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
15 simp12 1203 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
16 simp2 1136 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
17 simp3 1137 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 22718 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))
1918fveq2d 6910 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))))
2019fveq1d 6908 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾))
21 fvif 6922 . . . . . . 7 (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))
2221fveq1i 6907 . . . . . 6 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾)
23 iffv 6923 . . . . . 6 (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
2422, 23eqtri 2762 . . . . 5 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
25 eqid 2734 . . . . . . . . . . . . 13 (var1𝑅) = (var1𝑅)
26 eqid 2734 . . . . . . . . . . . . 13 (mulGrp‘𝑃) = (mulGrp‘𝑃)
27 eqid 2734 . . . . . . . . . . . . 13 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
281, 25, 26, 27ply1idvr1 22313 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
29283ad2ant2 1133 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
3029eqcomd 2740 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
3130fveq2d 6910 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
3231fveq1d 6908 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅)))‘𝐾))
331ply1lmod 22268 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34333ad2ant2 1133 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35 0nn0 12538 . . . . . . . . . . . . . 14 0 ∈ ℕ0
36 eqid 2734 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
371, 25, 26, 27, 36ply1moncl 22289 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
3835, 37mpan2 691 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
39383ad2ant2 1133 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
40 eqid 2734 . . . . . . . . . . . . 13 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2734 . . . . . . . . . . . . 13 ( ·𝑠𝑃) = ( ·𝑠𝑃)
42 eqid 2734 . . . . . . . . . . . . 13 (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
4336, 40, 41, 42lmodvs1 20904 . . . . . . . . . . . 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 2740 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
4645fveq2d 6910 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
4746fveq1d 6908 . . . . . . . 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 1136 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
491ply1sca 22269 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50493ad2ant2 1133 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
5150eqcomd 2740 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
5251fveq2d 6910 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
53 eqid 2734 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2734 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
5553, 54ringidcl 20279 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
56553ad2ant2 1133 . . . . . . . . . . 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 2734 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
6059, 53, 1, 25, 41, 26, 27coe1tm 22291 . . . . . . . . . 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 1370 . . . . . . . . 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 2738 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
6362ifbid 4553 . . . . . . . . . 10 (𝑘 = 𝐾 → if(𝑘 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
6463adantl 481 . . . . . . . . 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 6919 . . . . . . . . . . 11 (1r‘(Scalar‘𝑃)) ∈ V
66 fvex 6919 . . . . . . . . . . 11 (0g𝑅) ∈ V
6765, 66ifex 4580 . . . . . . . . . 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 7022 . . . . . . . 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 2778 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
711, 12, 59coe1z 22281 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
72713ad2ant2 1133 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
7372fveq1d 6908 . . . . . . . 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 7221 . . . . . . . . 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 2774 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = (0g𝑅))
7870, 77ifeq12d 4551 . . . . . 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 1132 . . . . 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 2786 . . . 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 2774 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8281mpoeq3dva 7509 . 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 481 . . . . . . . . 9 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 𝑅 = (Scalar‘𝑃))
8483eqcomd 2740 . . . . . . . 8 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
8584fveq2d 6910 . . . . . . 7 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
8685ifeq1d 4549 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
8786mpoeq3dv 7511 . . . . 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 4536 . . . . . . . 8 (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (1r‘(Scalar‘𝑃)))
8988ifeq1d 4549 . . . . . . 7 (𝐾 = 0 → if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)) = if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)))
9089adantr 480 . . . . . 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 7511 . . . . 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 22468 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9592, 94eqtrid 2786 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
96953adant3 1131 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9796adantl 481 . . . . 5 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9887, 91, 973eqtr4d 2784 . . . 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 4536 . . . . . 6 (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
10099eqcomd 2740 . . . . 5 (𝐾 = 0 → 1 = if(𝐾 = 0, 1 , 0 ))
101100adantr 480 . . . 4 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 1 = if(𝐾 = 0, 1 , 0 ))
10298, 101eqtrd 2774 . . 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 4570 . . . . . . 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 7511 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
106 iffalse 4539 . . . . . . . 8 𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (0g𝑅))
107106adantr 480 . . . . . . 7 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (0g𝑅))
108107ifeq1d 4549 . . . . . 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 7511 . . . . 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 1147 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
111110adantl 481 . . . . . 6 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
112 decpmatid.0 . . . . . . 7 0 = (0g𝐴)
11393, 59mat0op 22440 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
114112, 113eqtrid 2786 . . . . . 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 2784 . . . 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 4539 . . . . . 6 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118117eqcomd 2740 . . . . 5 𝐾 = 0 → 0 = if(𝐾 = 0, 1 , 0 ))
119118adantr 480 . . . 4 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → 0 = if(𝐾 = 0, 1 , 0 ))
120116, 119eqtrd 2774 . . 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 812 . 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 2778 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 1086   = wceq 1536  wcel 2105  Vcvv 3477  ifcif 4530  {csn 4630  cmpt 5230   × cxp 5686  cfv 6562  (class class class)co 7430  cmpo 7432  Fincfn 8983  0cc0 11152  0cn0 12523  Basecbs 17244  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485  .gcmg 19097  mulGrpcmgp 20151  1rcur 20198  Ringcrg 20250  LModclmod 20874  var1cv1 22192  Poly1cpl1 22193  coe1cco1 22194   Mat cmat 22426   decompPMat cdecpmat 22783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-dsmm 21769  df-frlm 21784  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-mamu 22410  df-mat 22427  df-decpmat 22784
This theorem is referenced by:  idpm2idmp  22822
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