Metamath Proof Explorer


Theorem pm2mp

Description: The transformation of a sum of matrices having scaled monomials with the same power as entries into a sum of scaled monomials as a polynomial over matrices. (Contributed by AV, 12-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 pm2mp N Fin R CRing M K 0 finSupp 0 A M I C n 0 n E Y · ˙ T M n = Q n 0 M n ˙ n × ˙ 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 eqid 0 C = 0 C
16 crngring R CRing R Ring
17 16 anim2i N Fin R CRing N Fin R Ring
18 1 2 pmatring N Fin R Ring C Ring
19 ringcmn C Ring C CMnd
20 17 18 19 3syl N Fin R CRing C CMnd
21 20 adantr N Fin R CRing M K 0 finSupp 0 A M C CMnd
22 7 matring N Fin R Ring A Ring
23 16 22 sylan2 N Fin R CRing A Ring
24 9 ply1ring A Ring Q Ring
25 ringmnd Q Ring Q Mnd
26 23 24 25 3syl N Fin R CRing Q Mnd
27 26 adantr N Fin R CRing M K 0 finSupp 0 A M Q Mnd
28 nn0ex 0 V
29 28 a1i N Fin R CRing M K 0 finSupp 0 A M 0 V
30 eqid Base Q = Base Q
31 1 2 3 4 5 6 7 9 30 10 pm2mpghm N Fin R Ring I C GrpHom Q
32 16 31 sylan2 N Fin R CRing I C GrpHom Q
33 32 adantr N Fin R CRing M K 0 finSupp 0 A M I C GrpHom Q
34 ghmmhm I C GrpHom Q I C MndHom Q
35 33 34 syl N Fin R CRing M K 0 finSupp 0 A M I C MndHom Q
36 17 adantr N Fin R CRing M K 0 finSupp 0 A M N Fin R Ring
37 36 adantr N Fin R CRing M K 0 finSupp 0 A M n 0 N Fin R Ring
38 elmapi M K 0 M : 0 K
39 38 adantr M K 0 finSupp 0 A M M : 0 K
40 39 adantl N Fin R CRing M K 0 finSupp 0 A M M : 0 K
41 40 ffvelcdmda N Fin R CRing M K 0 finSupp 0 A M n 0 M n K
42 simpr N Fin R CRing M K 0 finSupp 0 A M n 0 n 0
43 7 8 14 1 2 3 13 11 12 mat2pmatscmxcl N Fin R Ring M n K n 0 n E Y · ˙ T M n B
44 37 41 42 43 syl12anc N Fin R CRing M K 0 finSupp 0 A M n 0 n E Y · ˙ T M n B
45 fvexd N Fin R CRing M K 0 finSupp 0 A M 0 C V
46 ovexd N Fin R CRing M K 0 finSupp 0 A M n 0 n E Y · ˙ T M n V
47 simpr N Fin R CRing M K 0 M K 0
48 fvex 0 A V
49 fsuppmapnn0ub M K 0 0 A V finSupp 0 A M y 0 x 0 y < x M x = 0 A
50 47 48 49 sylancl N Fin R CRing M K 0 finSupp 0 A M y 0 x 0 y < x M x = 0 A
51 csbov12g x 0 x / n n E Y · ˙ T M n = x / n n E Y · ˙ x / n T M n
52 csbov1g x 0 x / n n E Y = x / n n E Y
53 csbvarg x 0 x / n n = x
54 53 oveq1d x 0 x / n n E Y = x E Y
55 52 54 eqtrd x 0 x / n n E Y = x E Y
56 csbfv2g x 0 x / n T M n = T x / n M n
57 csbfv2g x 0 x / n M n = M x / n n
58 53 fveq2d x 0 M x / n n = M x
59 57 58 eqtrd x 0 x / n M n = M x
60 59 fveq2d x 0 T x / n M n = T M x
61 56 60 eqtrd x 0 x / n T M n = T M x
62 55 61 oveq12d x 0 x / n n E Y · ˙ x / n T M n = x E Y · ˙ T M x
63 51 62 eqtrd x 0 x / n n E Y · ˙ T M n = x E Y · ˙ T M x
64 63 adantl N Fin R CRing M K 0 y 0 x 0 x / n n E Y · ˙ T M n = x E Y · ˙ T M x
65 64 adantr N Fin R CRing M K 0 y 0 x 0 M x = 0 A x / n n E Y · ˙ T M n = x E Y · ˙ T M x
66 fveq2 M x = 0 A T M x = T 0 A
67 66 oveq2d M x = 0 A x E Y · ˙ T M x = x E Y · ˙ T 0 A
68 14 7 8 1 2 3 mat2pmatghm N Fin R Ring T A GrpHom C
69 16 68 sylan2 N Fin R CRing T A GrpHom C
70 69 ad3antrrr N Fin R CRing M K 0 y 0 x 0 T A GrpHom C
71 ghmmhm T A GrpHom C T A MndHom C
72 eqid 0 A = 0 A
73 72 15 mhm0 T A MndHom C T 0 A = 0 C
74 70 71 73 3syl N Fin R CRing M K 0 y 0 x 0 T 0 A = 0 C
75 74 oveq2d N Fin R CRing M K 0 y 0 x 0 x E Y · ˙ T 0 A = x E Y · ˙ 0 C
76 1 ply1ring R Ring P Ring
77 16 76 syl R CRing P Ring
78 2 matlmod N Fin P Ring C LMod
79 77 78 sylan2 N Fin R CRing C LMod
80 79 ad3antrrr N Fin R CRing M K 0 y 0 x 0 C LMod
81 eqid mulGrp P = mulGrp P
82 eqid Base P = Base P
83 81 82 mgpbas Base P = Base mulGrp P
84 77 adantl N Fin R CRing P Ring
85 81 ringmgp P Ring mulGrp P Mnd
86 84 85 syl N Fin R CRing mulGrp P Mnd
87 86 ad3antrrr N Fin R CRing M K 0 y 0 x 0 mulGrp P Mnd
88 simpr N Fin R CRing M K 0 y 0 x 0 x 0
89 16 adantl N Fin R CRing R Ring
90 12 1 82 vr1cl R Ring Y Base P
91 89 90 syl N Fin R CRing Y Base P
92 91 ad3antrrr N Fin R CRing M K 0 y 0 x 0 Y Base P
93 83 11 87 88 92 mulgnn0cld N Fin R CRing M K 0 y 0 x 0 x E Y Base P
94 1 ply1crng R CRing P CRing
95 2 matsca2 N Fin P CRing P = Scalar C
96 94 95 sylan2 N Fin R CRing P = Scalar C
97 96 eqcomd N Fin R CRing Scalar C = P
98 97 ad3antrrr N Fin R CRing M K 0 y 0 x 0 Scalar C = P
99 98 fveq2d N Fin R CRing M K 0 y 0 x 0 Base Scalar C = Base P
100 93 99 eleqtrrd N Fin R CRing M K 0 y 0 x 0 x E Y Base Scalar C
101 eqid Scalar C = Scalar C
102 eqid Base Scalar C = Base Scalar C
103 101 13 102 15 lmodvs0 C LMod x E Y Base Scalar C x E Y · ˙ 0 C = 0 C
104 80 100 103 syl2anc N Fin R CRing M K 0 y 0 x 0 x E Y · ˙ 0 C = 0 C
105 75 104 eqtrd N Fin R CRing M K 0 y 0 x 0 x E Y · ˙ T 0 A = 0 C
106 67 105 sylan9eqr N Fin R CRing M K 0 y 0 x 0 M x = 0 A x E Y · ˙ T M x = 0 C
107 65 106 eqtrd N Fin R CRing M K 0 y 0 x 0 M x = 0 A x / n n E Y · ˙ T M n = 0 C
108 107 ex N Fin R CRing M K 0 y 0 x 0 M x = 0 A x / n n E Y · ˙ T M n = 0 C
109 108 imim2d N Fin R CRing M K 0 y 0 x 0 y < x M x = 0 A y < x x / n n E Y · ˙ T M n = 0 C
110 109 ralimdva N Fin R CRing M K 0 y 0 x 0 y < x M x = 0 A x 0 y < x x / n n E Y · ˙ T M n = 0 C
111 110 reximdva N Fin R CRing M K 0 y 0 x 0 y < x M x = 0 A y 0 x 0 y < x x / n n E Y · ˙ T M n = 0 C
112 50 111 syld N Fin R CRing M K 0 finSupp 0 A M y 0 x 0 y < x x / n n E Y · ˙ T M n = 0 C
113 112 impr N Fin R CRing M K 0 finSupp 0 A M y 0 x 0 y < x x / n n E Y · ˙ T M n = 0 C
114 45 46 113 mptnn0fsupp N Fin R CRing M K 0 finSupp 0 A M finSupp 0 C n 0 n E Y · ˙ T M n
115 3 15 21 27 29 35 44 114 gsummptmhm N Fin R CRing M K 0 finSupp 0 A M Q n 0 I n E Y · ˙ T M n = I C n 0 n E Y · ˙ T M n
116 simpll N Fin R CRing M K 0 finSupp 0 A M n 0 N Fin R CRing
117 1 2 3 4 5 6 7 8 9 10 11 12 13 14 monmat2matmon N Fin R CRing M n K n 0 I n E Y · ˙ T M n = M n ˙ n × ˙ X
118 116 41 42 117 syl12anc N Fin R CRing M K 0 finSupp 0 A M n 0 I n E Y · ˙ T M n = M n ˙ n × ˙ X
119 118 mpteq2dva N Fin R CRing M K 0 finSupp 0 A M n 0 I n E Y · ˙ T M n = n 0 M n ˙ n × ˙ X
120 119 oveq2d N Fin R CRing M K 0 finSupp 0 A M Q n 0 I n E Y · ˙ T M n = Q n 0 M n ˙ n × ˙ X
121 115 120 eqtr3d N Fin R CRing M K 0 finSupp 0 A M I C n 0 n E Y · ˙ T M n = Q n 0 M n ˙ n × ˙ X