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Theorem decpmatid 20623
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 20546 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
433adant3 1101 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐶 ∈ Ring)
5 eqid 2651 . . . . 5 (Base‘𝐶) = (Base‘𝐶)
6 decpmatid.i . . . . 5 𝐼 = (1r𝐶)
75, 6ringidcl 18614 . . . 4 (𝐶 ∈ Ring → 𝐼 ∈ (Base‘𝐶))
84, 7syl 17 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐼 ∈ (Base‘𝐶))
9 simp3 1083 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝐾 ∈ ℕ0)
102, 5decpmatval 20618 . . 3 ((𝐼 ∈ (Base‘𝐶) ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
118, 9, 10syl2anc 694 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (𝐼 decompPMat 𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝐼𝑗))‘𝐾)))
12 eqid 2651 . . . . . . 7 (0g𝑃) = (0g𝑃)
13 eqid 2651 . . . . . . 7 (1r𝑃) = (1r𝑃)
14 simp11 1111 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑁 ∈ Fin)
15 simp12 1112 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑅 ∈ Ring)
16 simp2 1082 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
17 simp3 1083 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
181, 2, 12, 13, 14, 15, 16, 17, 6pmat1ovd 20550 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝐼𝑗) = if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))
1918fveq2d 6233 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (coe1‘(𝑖𝐼𝑗)) = (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))))
2019fveq1d 6231 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾))
21 fvif 6242 . . . . . . 7 (coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃))) = if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))
2221fveq1i 6230 . . . . . 6 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾)
23 iffv 6243 . . . . . 6 (if(𝑖 = 𝑗, (coe1‘(1r𝑃)), (coe1‘(0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
2422, 23eqtri 2673 . . . . 5 ((coe1‘if(𝑖 = 𝑗, (1r𝑃), (0g𝑃)))‘𝐾) = if(𝑖 = 𝑗, ((coe1‘(1r𝑃))‘𝐾), ((coe1‘(0g𝑃))‘𝐾))
25 eqid 2651 . . . . . . . . . . . . 13 (var1𝑅) = (var1𝑅)
26 eqid 2651 . . . . . . . . . . . . 13 (mulGrp‘𝑃) = (mulGrp‘𝑃)
27 eqid 2651 . . . . . . . . . . . . 13 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
281, 25, 26, 27ply1idvr1 19711 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
29283ad2ant2 1103 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = (1r𝑃))
3029eqcomd 2657 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑃) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
3130fveq2d 6233 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(1r𝑃)) = (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
3231fveq1d 6231 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = ((coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅)))‘𝐾))
331ply1lmod 19670 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → 𝑃 ∈ LMod)
34333ad2ant2 1103 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑃 ∈ LMod)
35 0nn0 11345 . . . . . . . . . . . . . 14 0 ∈ ℕ0
36 eqid 2651 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
371, 25, 26, 27, 36ply1moncl 19689 . . . . . . . . . . . . . 14 ((𝑅 ∈ Ring ∧ 0 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
3835, 37mpan2 707 . . . . . . . . . . . . 13 (𝑅 ∈ Ring → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
39383ad2ant2 1103 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
40 eqid 2651 . . . . . . . . . . . . 13 (Scalar‘𝑃) = (Scalar‘𝑃)
41 eqid 2651 . . . . . . . . . . . . 13 ( ·𝑠𝑃) = ( ·𝑠𝑃)
42 eqid 2651 . . . . . . . . . . . . 13 (1r‘(Scalar‘𝑃)) = (1r‘(Scalar‘𝑃))
4336, 40, 41, 42lmodvs1 18939 . . . . . . . . . . . 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 694 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0(.g‘(mulGrp‘𝑃))(var1𝑅)))
4544eqcomd 2657 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (0(.g‘(mulGrp‘𝑃))(var1𝑅)) = ((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅))))
4645fveq2d 6233 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0(.g‘(mulGrp‘𝑃))(var1𝑅))) = (coe1‘((1r‘(Scalar‘𝑃))( ·𝑠𝑃)(0(.g‘(mulGrp‘𝑃))(var1𝑅)))))
4746fveq1d 6231 . . . . . . . 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 1082 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 ∈ Ring)
491ply1sca 19671 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
50493ad2ant2 1103 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → 𝑅 = (Scalar‘𝑃))
5150eqcomd 2657 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (Scalar‘𝑃) = 𝑅)
5251fveq2d 6233 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
53 eqid 2651 . . . . . . . . . . . . 13 (Base‘𝑅) = (Base‘𝑅)
54 eqid 2651 . . . . . . . . . . . . 13 (1r𝑅) = (1r𝑅)
5553, 54ringidcl 18614 . . . . . . . . . . . 12 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
56553ad2ant2 1103 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (1r𝑅) ∈ (Base‘𝑅))
5752, 56eqeltrd 2730 . . . . . . . . . 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 2651 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
6059, 53, 1, 25, 41, 26, 27coe1tm 19691 . . . . . . . . . 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 1366 . . . . . . . . 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 2655 . . . . . . . . . . 11 (𝑘 = 𝐾 → (𝑘 = 0 ↔ 𝐾 = 0))
6362ifbid 4141 . . . . . . . . . 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 6239 . . . . . . . . . . 11 (1r‘(Scalar‘𝑃)) ∈ V
66 fvex 6239 . . . . . . . . . . 11 (0g𝑅) ∈ V
6765, 66ifex 4189 . . . . . . . . . 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 6327 . . . . . . . 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 2689 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(1r𝑃))‘𝐾) = if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)))
711, 12, 59coe1z 19681 . . . . . . . . . 10 (𝑅 ∈ Ring → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
72713ad2ant2 1103 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → (coe1‘(0g𝑃)) = (ℕ0 × {(0g𝑅)}))
7372fveq1d 6231 . . . . . . . 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 6508 . . . . . . . . 9 (((0g𝑅) ∈ V ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7674, 9, 75syl2anc 694 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((ℕ0 × {(0g𝑅)})‘𝐾) = (0g𝑅))
7773, 76eqtrd 2685 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) → ((coe1‘(0g𝑃))‘𝐾) = (0g𝑅))
7870, 77ifeq12d 4139 . . . . . 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 1102 . . . . 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 2697 . . . 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 2685 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝐼𝑗))‘𝐾) = if(𝑖 = 𝑗, if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)), (0g𝑅)))
8281mpt2eq3dva 6761 . 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 2657 . . . . . . . 8 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (Scalar‘𝑃) = 𝑅)
8584fveq2d 6233 . . . . . . 7 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (1r‘(Scalar‘𝑃)) = (1r𝑅))
8685ifeq1d 4137 . . . . . 6 ((𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → if(𝑖 = 𝑗, (1r‘(Scalar‘𝑃)), (0g𝑅)) = if(𝑖 = 𝑗, (1r𝑅), (0g𝑅)))
8786mpt2eq3dv 6763 . . . . 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 4125 . . . . . . . 8 (𝐾 = 0 → if(𝐾 = 0, (1r‘(Scalar‘𝑃)), (0g𝑅)) = (1r‘(Scalar‘𝑃)))
8988ifeq1d 4137 . . . . . . 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 6763 . . . . 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 20301 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
9592, 94syl5eq 2697 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 1 = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (1r𝑅), (0g𝑅))))
96953adant3 1101 . . . . . 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 2695 . . . 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 4125 . . . . . 6 (𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 1 )
10099eqcomd 2657 . . . . 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 2685 . . 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 4158 . . . . . . 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 6763 . . . . 5 ((¬ 𝐾 = 0 ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐾 ∈ ℕ0)) → (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, (0g𝑅), (0g𝑅))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
106 iffalse 4128 . . . . . . . 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 4137 . . . . . 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 6763 . . . . 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 1078 . . . . . . 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 20273 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
114112, 113syl5eq 2697 . . . . . 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 2695 . . . 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 4128 . . . . . 6 𝐾 = 0 → if(𝐾 = 0, 1 , 0 ) = 0 )
118117eqcomd 2657 . . . . 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 2685 . . 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 848 . 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 2689 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 1054   = wceq 1523  wcel 2030  Vcvv 3231  ifcif 4119  {csn 4210  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692  Fincfn 7997  0cc0 9974  0cn0 11330  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  .gcmg 17587  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593  LModclmod 18911  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596   Mat cmat 20261   decompPMat cdecpmat 20615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-ascl 19362  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-decpmat 20616
This theorem is referenced by:  idpm2idmp  20654
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