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Theorem decpmatid 20394
 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 20317 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
433adant3 1074 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5 eqid 2610 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 decpmatid.i . . . . 5 𝐼 = (1r𝐶)
75, 6ringidcl 18391 . . . 4 (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
84, 7syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9 simp3 1056 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
102, 5decpmatval 20389 . . 3 ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
118, 9, 10syl2anc 691 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12 eqid 2610 . . . . . . 7 (0g𝑃) = (0g𝑃)
13 eqid 2610 . . . . . . 7 (1r𝑃) = (1r𝑃)
14 simp11 1084 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
15 simp12 1085 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
16 simp2 1055 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
17 simp3 1056 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 20321 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))
1918fveq2d 6107 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))))
2019fveq1d 6105 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾))
21 fvif 6114 . . . . . . 7 (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))
2221fveq1i 6104 . . . . . 6 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾)
23 iffv 6115 . . . . . 6 (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
2422, 23eqtri 2632 . . . . 5 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
25 eqid 2610 . . . . . . . . . . . . 13 (var1𝑅) = (var1𝑅)
26 eqid 2610 . . . . . . . . . . . . 13 (mulGrp‘𝑃) = (mulGrp‘𝑃)
27 eqid 2610 . . . . . . . . . . . . 13 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
281, 25, 26, 27ply1idvr1 19484 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
29283ad2ant2 1076 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
3029eqcomd 2616 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
3130fveq2d 6107 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
3231fveq1d 6105 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅)))‘𝐾))
331ply1lmod 19443 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34333ad2ant2 1076 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35 0nn0 11184 . . . . . . . . . . . . . 14 0 ∈ ℕ0
36 eqid 2610 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
371, 25, 26, 27, 36ply1moncl 19462 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
3835, 37mpan2 703 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
39383ad2ant2 1076 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
40 eqid 2610 . . . . . . . . . . . . 13 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2610 . . . . . . . . . . . . 13 ( ·𝑠𝑃) = ( ·𝑠𝑃)
42 eqid 2610 . . . . . . . . . . . . 13 (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
4336, 40, 41, 42lmodvs1 18714 . . . . . . . . . . . 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 691 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
4544eqcomd 2616 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
4645fveq2d 6107 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
4746fveq1d 6105 . . . . . . . 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 1055 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
491ply1sca 19444 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50493ad2ant2 1076 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
5150eqcomd 2616 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
5251fveq2d 6107 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
53 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2610 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
5553, 54ringidcl 18391 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
56553ad2ant2 1076 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑅) ∈ (Base‘𝑅))
5752, 56eqeltrd 2688 . . . . . . . . . 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 2610 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
6059, 53, 1, 25, 41, 26, 27coe1tm 19464 . . . . . . . . . 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 1318 . . . . . . . . 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 2614 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
6362ifbid 4058 . . . . . . . . . 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 6113 . . . . . . . . . . 11 (1r‘(Scalar‘𝑃)) ∈ V
66 fvex 6113 . . . . . . . . . . 11 (0g𝑅) ∈ V
6765, 66ifex 4106 . . . . . . . . . 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 6197 . . . . . . . 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 2648 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
711, 12, 59coe1z 19454 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
72713ad2ant2 1076 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
7372fveq1d 6105 . . . . . . . 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 6372 . . . . . . . . 9 (((0g𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7674, 9, 75syl2anc 691 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7773, 76eqtrd 2644 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = (0g𝑅))
7870, 77ifeq12d 4056 . . . . . 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 1075 . . . . 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, 79syl5eq 2656 . . . 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 2644 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8281mpt2eq3dva 6617 . 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 2616 . . . . . . . 8 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
8584fveq2d 6107 . . . . . . 7 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
8685ifeq1d 4054 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
8786mpt2eq3dv 6619 . . . . 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 4042 . . . . . . . 8 (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (1r‘(Scalar‘𝑃)))
8988ifeq1d 4054 . . . . . . 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)))
9190mpt2eq3dv 6619 . . . . 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 20072 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9592, 94syl5eq 2656 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
96953adant3 1074 . . . . . 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 2654 . . . 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 4042 . . . . . 6 (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
10099eqcomd 2616 . . . . 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 2644 . . 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 4075 . . . . . . 7 if(𝑖 = 𝑗, (0g𝑅), (0g𝑅)) = (0g𝑅)
104103a1i 11 . . . . . 6 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (0g𝑅), (0g𝑅)) = (0g𝑅))
105104mpt2eq3dv 6619 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
106 iffalse 4045 . . . . . . . 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 4054 . . . . . 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)))
109108mpt2eq3dv 6619 . . . . 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 1051 . . . . . . 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 20044 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
114112, 113syl5eq 2656 . . . . . 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 2654 . . . 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 4045 . . . . . 6 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118117eqcomd 2616 . . . . 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 2644 . . 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 827 . 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 2648 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036  {csn 4125   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  0cc0 9815  ℕ0cn0 11169  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  .gcmg 17363  mulGrpcmgp 18312  1rcur 18324  Ringcrg 18370  LModclmod 18686  var1cv1 19367  Poly1cpl1 19368  coe1cco1 19369   Mat cmat 20032   decompPMat cdecpmat 20386 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-ofr 6796  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-sra 18993  df-rgmod 18994  df-ascl 19135  df-psr 19177  df-mvr 19178  df-mpl 19179  df-opsr 19181  df-psr1 19371  df-vr1 19372  df-ply1 19373  df-coe1 19374  df-dsmm 19895  df-frlm 19910  df-mamu 20009  df-mat 20033  df-decpmat 20387 This theorem is referenced by:  idpm2idmp  20425
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