Metamath Proof Explorer


Theorem monmat2matmon

Description: The transformation of a polynomial matrix having scaled monomials with the same power as entries into a scaled monomial as a polynomial over matrices. (Contributed by AV, 11-Nov-2019) (Revised by AV, 7-Dec-2019)

Ref Expression
Hypotheses monmat2matmon.p P = Poly 1 R
monmat2matmon.c C = N Mat P
monmat2matmon.b B = Base C
monmat2matmon.m1 ˙ = Q
monmat2matmon.e1 × ˙ = mulGrp Q
monmat2matmon.x X = var 1 A
monmat2matmon.a A = N Mat R
monmat2matmon.k K = Base A
monmat2matmon.q Q = Poly 1 A
monmat2matmon.i I = N pMatToMatPoly R
monmat2matmon.e2 E = mulGrp P
monmat2matmon.y Y = var 1 R
monmat2matmon.m2 · ˙ = C
monmat2matmon.t T = N matToPolyMat R
Assertion monmat2matmon N Fin R CRing M K L 0 I L E Y · ˙ T M = M ˙ L × ˙ X

Proof

Step Hyp Ref Expression
1 monmat2matmon.p P = Poly 1 R
2 monmat2matmon.c C = N Mat P
3 monmat2matmon.b B = Base C
4 monmat2matmon.m1 ˙ = Q
5 monmat2matmon.e1 × ˙ = mulGrp Q
6 monmat2matmon.x X = var 1 A
7 monmat2matmon.a A = N Mat R
8 monmat2matmon.k K = Base A
9 monmat2matmon.q Q = Poly 1 A
10 monmat2matmon.i I = N pMatToMatPoly R
11 monmat2matmon.e2 E = mulGrp P
12 monmat2matmon.y Y = var 1 R
13 monmat2matmon.m2 · ˙ = C
14 monmat2matmon.t T = N matToPolyMat R
15 crngring R CRing R Ring
16 simpll N Fin R Ring M K L 0 N Fin
17 simplr N Fin R Ring M K L 0 R Ring
18 7 8 14 1 2 3 13 11 12 mat2pmatscmxcl N Fin R Ring M K L 0 L E Y · ˙ T M B
19 1 2 3 4 5 6 7 9 10 pm2mpfval N Fin R Ring L E Y · ˙ T M B I L E Y · ˙ T M = Q k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X
20 16 17 18 19 syl3anc N Fin R Ring M K L 0 I L E Y · ˙ T M = Q k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X
21 15 20 sylanl2 N Fin R CRing M K L 0 I L E Y · ˙ T M = Q k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X
22 simpll N Fin R CRing M K L 0 k 0 N Fin R CRing
23 simpr N Fin R CRing M K L 0 M K L 0
24 23 anim1i N Fin R CRing M K L 0 k 0 M K L 0 k 0
25 df-3an M K L 0 k 0 M K L 0 k 0
26 24 25 sylibr N Fin R CRing M K L 0 k 0 M K L 0 k 0
27 eqid 0 A = 0 A
28 1 2 7 8 27 11 12 13 14 monmatcollpw N Fin R CRing M K L 0 k 0 L E Y · ˙ T M decompPMat k = if k = L M 0 A
29 22 26 28 syl2anc N Fin R CRing M K L 0 k 0 L E Y · ˙ T M decompPMat k = if k = L M 0 A
30 29 oveq1d N Fin R CRing M K L 0 k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X = if k = L M 0 A ˙ k × ˙ X
31 15 a1i k 0 R CRing R Ring
32 31 anim2d k 0 N Fin R CRing N Fin R Ring
33 32 anim1d k 0 N Fin R CRing M K L 0 N Fin R Ring M K L 0
34 33 imdistanri N Fin R CRing M K L 0 k 0 N Fin R Ring M K L 0 k 0
35 ovif if k = L M 0 A ˙ k × ˙ X = if k = L M ˙ k × ˙ X 0 A ˙ k × ˙ X
36 7 matring N Fin R Ring A Ring
37 9 ply1sca A Ring A = Scalar Q
38 36 37 syl N Fin R Ring A = Scalar Q
39 38 ad2antrr N Fin R Ring M K L 0 k 0 A = Scalar Q
40 39 fveq2d N Fin R Ring M K L 0 k 0 0 A = 0 Scalar Q
41 40 oveq1d N Fin R Ring M K L 0 k 0 0 A ˙ k × ˙ X = 0 Scalar Q ˙ k × ˙ X
42 9 ply1lmod A Ring Q LMod
43 36 42 syl N Fin R Ring Q LMod
44 43 ad2antrr N Fin R Ring M K L 0 k 0 Q LMod
45 eqid mulGrp Q = mulGrp Q
46 eqid Base Q = Base Q
47 45 46 mgpbas Base Q = Base mulGrp Q
48 9 ply1ring A Ring Q Ring
49 36 48 syl N Fin R Ring Q Ring
50 45 ringmgp Q Ring mulGrp Q Mnd
51 49 50 syl N Fin R Ring mulGrp Q Mnd
52 51 ad2antrr N Fin R Ring M K L 0 k 0 mulGrp Q Mnd
53 simpr N Fin R Ring M K L 0 k 0 k 0
54 6 9 46 vr1cl A Ring X Base Q
55 36 54 syl N Fin R Ring X Base Q
56 55 ad2antrr N Fin R Ring M K L 0 k 0 X Base Q
57 47 5 52 53 56 mulgnn0cld N Fin R Ring M K L 0 k 0 k × ˙ X Base Q
58 eqid Scalar Q = Scalar Q
59 eqid 0 Scalar Q = 0 Scalar Q
60 eqid 0 Q = 0 Q
61 46 58 4 59 60 lmod0vs Q LMod k × ˙ X Base Q 0 Scalar Q ˙ k × ˙ X = 0 Q
62 44 57 61 syl2anc N Fin R Ring M K L 0 k 0 0 Scalar Q ˙ k × ˙ X = 0 Q
63 41 62 eqtrd N Fin R Ring M K L 0 k 0 0 A ˙ k × ˙ X = 0 Q
64 63 ifeq2d N Fin R Ring M K L 0 k 0 if k = L M ˙ k × ˙ X 0 A ˙ k × ˙ X = if k = L M ˙ k × ˙ X 0 Q
65 35 64 eqtrid N Fin R Ring M K L 0 k 0 if k = L M 0 A ˙ k × ˙ X = if k = L M ˙ k × ˙ X 0 Q
66 34 65 syl N Fin R CRing M K L 0 k 0 if k = L M 0 A ˙ k × ˙ X = if k = L M ˙ k × ˙ X 0 Q
67 30 66 eqtrd N Fin R CRing M K L 0 k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X = if k = L M ˙ k × ˙ X 0 Q
68 67 mpteq2dva N Fin R CRing M K L 0 k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X = k 0 if k = L M ˙ k × ˙ X 0 Q
69 68 oveq2d N Fin R CRing M K L 0 Q k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X = Q k 0 if k = L M ˙ k × ˙ X 0 Q
70 ringmnd Q Ring Q Mnd
71 49 70 syl N Fin R Ring Q Mnd
72 71 adantr N Fin R Ring M K L 0 Q Mnd
73 nn0ex 0 V
74 73 a1i N Fin R Ring M K L 0 0 V
75 simprr N Fin R Ring M K L 0 L 0
76 eqid k 0 if k = L M ˙ k × ˙ X 0 Q = k 0 if k = L M ˙ k × ˙ X 0 Q
77 38 fveq2d N Fin R Ring Base A = Base Scalar Q
78 8 77 eqtrid N Fin R Ring K = Base Scalar Q
79 78 eleq2d N Fin R Ring M K M Base Scalar Q
80 79 biimpcd M K N Fin R Ring M Base Scalar Q
81 80 adantr M K L 0 N Fin R Ring M Base Scalar Q
82 81 impcom N Fin R Ring M K L 0 M Base Scalar Q
83 82 adantr N Fin R Ring M K L 0 k 0 M Base Scalar Q
84 eqid Base Scalar Q = Base Scalar Q
85 46 58 4 84 lmodvscl Q LMod M Base Scalar Q k × ˙ X Base Q M ˙ k × ˙ X Base Q
86 44 83 57 85 syl3anc N Fin R Ring M K L 0 k 0 M ˙ k × ˙ X Base Q
87 86 ralrimiva N Fin R Ring M K L 0 k 0 M ˙ k × ˙ X Base Q
88 60 72 74 75 76 87 gsummpt1n0 N Fin R Ring M K L 0 Q k 0 if k = L M ˙ k × ˙ X 0 Q = L / k M ˙ k × ˙ X
89 15 88 sylanl2 N Fin R CRing M K L 0 Q k 0 if k = L M ˙ k × ˙ X 0 Q = L / k M ˙ k × ˙ X
90 69 89 eqtrd N Fin R CRing M K L 0 Q k 0 L E Y · ˙ T M decompPMat k ˙ k × ˙ X = L / k M ˙ k × ˙ X
91 csbov2g L 0 L / k M ˙ k × ˙ X = M ˙ L / k k × ˙ X
92 csbov1g L 0 L / k k × ˙ X = L / k k × ˙ X
93 csbvarg L 0 L / k k = L
94 93 oveq1d L 0 L / k k × ˙ X = L × ˙ X
95 92 94 eqtrd L 0 L / k k × ˙ X = L × ˙ X
96 95 oveq2d L 0 M ˙ L / k k × ˙ X = M ˙ L × ˙ X
97 91 96 eqtrd L 0 L / k M ˙ k × ˙ X = M ˙ L × ˙ X
98 97 ad2antll N Fin R CRing M K L 0 L / k M ˙ k × ˙ X = M ˙ L × ˙ X
99 21 90 98 3eqtrd N Fin R CRing M K L 0 I L E Y · ˙ T M = M ˙ L × ˙ X