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Theorem decpmatid 22776
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 22698 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
433adant3 1133 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5 eqid 2737 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 decpmatid.i . . . . 5 𝐼 = (1r𝐶)
75, 6ringidcl 20262 . . . 4 (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
84, 7syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9 simp3 1139 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
102, 5decpmatval 22771 . . 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 2737 . . . . . . 7 (0g𝑃) = (0g𝑃)
13 eqid 2737 . . . . . . 7 (1r𝑃) = (1r𝑃)
14 simp11 1204 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
15 simp12 1205 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
16 simp2 1138 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
17 simp3 1139 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 22703 . . . . . 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 2765 . . . . 5 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
25 eqid 2737 . . . . . . . . . . . . 13 (var1𝑅) = (var1𝑅)
26 eqid 2737 . . . . . . . . . . . . 13 (mulGrp‘𝑃) = (mulGrp‘𝑃)
27 eqid 2737 . . . . . . . . . . . . 13 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
281, 25, 26, 27ply1idvr1 22298 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
29283ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
3029eqcomd 2743 . . . . . . . . . 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 22253 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34333ad2ant2 1135 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35 0nn0 12541 . . . . . . . . . . . . . 14 0 ∈ ℕ0
36 eqid 2737 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
371, 25, 26, 27, 36ply1moncl 22274 . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
40 eqid 2737 . . . . . . . . . . . . 13 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2737 . . . . . . . . . . . . 13 ( ·𝑠𝑃) = ( ·𝑠𝑃)
42 eqid 2737 . . . . . . . . . . . . 13 (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
4336, 40, 41, 42lmodvs1 20888 . . . . . . . . . . . 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 2743 . . . . . . . . . 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 1138 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
491ply1sca 22254 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50493ad2ant2 1135 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
5150eqcomd 2743 . . . . . . . . . . . 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 2737 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2737 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
5553, 54ringidcl 20262 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
56553ad2ant2 1135 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑅) ∈ (Base‘𝑅))
5752, 56eqeltrd 2841 . . . . . . . . . 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 2737 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
6059, 53, 1, 25, 41, 26, 27coe1tm 22276 . . . . . . . . . 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 1373 . . . . . . . . 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 2741 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
6362ifbid 4549 . . . . . . . . . 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 4576 . . . . . . . . . 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 7023 . . . . . . . 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 2781 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
711, 12, 59coe1z 22266 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
72713ad2ant2 1135 . . . . . . . . 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 7222 . . . . . . . . 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 2777 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = (0g𝑅))
7870, 77ifeq12d 4547 . . . . . 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 1134 . . . . 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 2789 . . . 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 2777 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8281mpoeq3dva 7510 . 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 2743 . . . . . . . 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 4545 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
8786mpoeq3dv 7512 . . . . 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 4531 . . . . . . . 8 (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (1r‘(Scalar‘𝑃)))
8988ifeq1d 4545 . . . . . . 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 7512 . . . . 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 22453 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9592, 94eqtrid 2789 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
96953adant3 1133 . . . . . 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 2787 . . . 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 4531 . . . . . 6 (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
10099eqcomd 2743 . . . . 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 2777 . . 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 4566 . . . . . . 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 7512 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
106 iffalse 4534 . . . . . . . 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 4545 . . . . . 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 7512 . . . . 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 1149 . . . . . . 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 22425 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
114112, 113eqtrid 2789 . . . . . 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 2787 . . . 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 4534 . . . . . 6 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118117eqcomd 2743 . . . . 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 2777 . . 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 2781 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 1087   = wceq 1540  wcel 2108  Vcvv 3480  ifcif 4525  {csn 4626  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  Fincfn 8985  0cc0 11155  0cn0 12526  Basecbs 17247  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484  .gcmg 19085  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  LModclmod 20858  var1cv1 22177  Poly1cpl1 22178  coe1cco1 22179   Mat cmat 22411   decompPMat cdecpmat 22768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  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 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-sra 21172  df-rgmod 21173  df-dsmm 21752  df-frlm 21767  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-mamu 22395  df-mat 22412  df-decpmat 22769
This theorem is referenced by:  idpm2idmp  22807
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