Metamath Proof Explorer


Theorem chfacfscmulcl

Description: Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019)

Ref Expression
Hypotheses chfacfisf.a A = N Mat R
chfacfisf.b B = Base A
chfacfisf.p P = Poly 1 R
chfacfisf.y Y = N Mat P
chfacfisf.r × ˙ = Y
chfacfisf.s - ˙ = - Y
chfacfisf.0 0 ˙ = 0 Y
chfacfisf.t T = N matToPolyMat R
chfacfisf.g G = n 0 if n = 0 0 ˙ - ˙ T M × ˙ T b 0 if n = s + 1 T b s if s + 1 < n 0 ˙ T b n 1 - ˙ T M × ˙ T b n
chfacfscmulcl.x X = var 1 R
chfacfscmulcl.m · ˙ = Y
chfacfscmulcl.e × ˙ = mulGrp P
Assertion chfacfscmulcl N Fin R CRing M B s b B 0 s K 0 K × ˙ X · ˙ G K Base Y

Proof

Step Hyp Ref Expression
1 chfacfisf.a A = N Mat R
2 chfacfisf.b B = Base A
3 chfacfisf.p P = Poly 1 R
4 chfacfisf.y Y = N Mat P
5 chfacfisf.r × ˙ = Y
6 chfacfisf.s - ˙ = - Y
7 chfacfisf.0 0 ˙ = 0 Y
8 chfacfisf.t T = N matToPolyMat R
9 chfacfisf.g G = n 0 if n = 0 0 ˙ - ˙ T M × ˙ T b 0 if n = s + 1 T b s if s + 1 < n 0 ˙ T b n 1 - ˙ T M × ˙ T b n
10 chfacfscmulcl.x X = var 1 R
11 chfacfscmulcl.m · ˙ = Y
12 chfacfscmulcl.e × ˙ = mulGrp P
13 crngring R CRing R Ring
14 3 4 pmatlmod N Fin R Ring Y LMod
15 13 14 sylan2 N Fin R CRing Y LMod
16 15 3adant3 N Fin R CRing M B Y LMod
17 16 3ad2ant1 N Fin R CRing M B s b B 0 s K 0 Y LMod
18 eqid mulGrp P = mulGrp P
19 eqid Base P = Base P
20 18 19 mgpbas Base P = Base mulGrp P
21 3 ply1ring R Ring P Ring
22 13 21 syl R CRing P Ring
23 22 3ad2ant2 N Fin R CRing M B P Ring
24 18 ringmgp P Ring mulGrp P Mnd
25 23 24 syl N Fin R CRing M B mulGrp P Mnd
26 25 3ad2ant1 N Fin R CRing M B s b B 0 s K 0 mulGrp P Mnd
27 simp3 N Fin R CRing M B s b B 0 s K 0 K 0
28 13 3ad2ant2 N Fin R CRing M B R Ring
29 10 3 19 vr1cl R Ring X Base P
30 28 29 syl N Fin R CRing M B X Base P
31 30 3ad2ant1 N Fin R CRing M B s b B 0 s K 0 X Base P
32 20 12 26 27 31 mulgnn0cld N Fin R CRing M B s b B 0 s K 0 K × ˙ X Base P
33 3 ply1crng R CRing P CRing
34 33 anim2i N Fin R CRing N Fin P CRing
35 34 3adant3 N Fin R CRing M B N Fin P CRing
36 4 matsca2 N Fin P CRing P = Scalar Y
37 35 36 syl N Fin R CRing M B P = Scalar Y
38 37 eqcomd N Fin R CRing M B Scalar Y = P
39 38 fveq2d N Fin R CRing M B Base Scalar Y = Base P
40 39 3ad2ant1 N Fin R CRing M B s b B 0 s K 0 Base Scalar Y = Base P
41 32 40 eleqtrrd N Fin R CRing M B s b B 0 s K 0 K × ˙ X Base Scalar Y
42 1 2 3 4 5 6 7 8 9 chfacfisf N Fin R Ring M B s b B 0 s G : 0 Base Y
43 13 42 syl3anl2 N Fin R CRing M B s b B 0 s G : 0 Base Y
44 43 3adant3 N Fin R CRing M B s b B 0 s K 0 G : 0 Base Y
45 44 27 ffvelcdmd N Fin R CRing M B s b B 0 s K 0 G K Base Y
46 eqid Base Y = Base Y
47 eqid Scalar Y = Scalar Y
48 eqid Base Scalar Y = Base Scalar Y
49 46 47 11 48 lmodvscl Y LMod K × ˙ X Base Scalar Y G K Base Y K × ˙ X · ˙ G K Base Y
50 17 41 45 49 syl3anc N Fin R CRing M B s b B 0 s K 0 K × ˙ X · ˙ G K Base Y