Metamath Proof Explorer

Theorem cpmidpmatlem2

Description: Lemma 2 for cpmidpmat . (Contributed by AV, 14-Nov-2019) (Proof shortened by AV, 7-Dec-2019)

Ref Expression
Hypotheses cpmidgsum.a 
|- A = ( N Mat R )
cpmidgsum.b 
|- B = ( Base ` A )
cpmidgsum.p 
|- P = ( Poly1 ` R )
cpmidgsum.y 
|- Y = ( N Mat P )
cpmidgsum.x 
|- X = ( var1 ` R )
cpmidgsum.e 
|- .^ = ( .g ` ( mulGrp ` P ) )
cpmidgsum.m 
|- .x. = ( .s ` Y )
cpmidgsum.1 
|- .1. = ( 1r ` Y )
cpmidgsum.u 
|- U = ( algSc ` P )
cpmidgsum.c 
|- C = ( N CharPlyMat R )
cpmidgsum.k 
|- K = ( C ` M )
cpmidgsum.h 
|- H = ( K .x. .1. )
cpmidgsumm2pm.o 
|- O = ( 1r ` A )
cpmidgsumm2pm.m 
|- .* = ( .s ` A )
cpmidgsumm2pm.t 
|- T = ( N matToPolyMat R )
cpmidpmat.g 
|- G = ( k e. NN0 |-> ( ( ( coe1 ` K ) ` k ) .* O ) )
Assertion cpmidpmatlem2 
|- ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> G e. ( B ^m NN0 ) )

Proof

Step Hyp Ref Expression
1 cpmidgsum.a 
 |-  A = ( N Mat R )
2 cpmidgsum.b 
 |-  B = ( Base ` A )
3 cpmidgsum.p 
 |-  P = ( Poly1 ` R )
4 cpmidgsum.y 
 |-  Y = ( N Mat P )
5 cpmidgsum.x 
 |-  X = ( var1 ` R )
6 cpmidgsum.e 
 |-  .^ = ( .g ` ( mulGrp ` P ) )
7 cpmidgsum.m 
 |-  .x. = ( .s ` Y )
8 cpmidgsum.1 
 |-  .1. = ( 1r ` Y )
9 cpmidgsum.u 
 |-  U = ( algSc ` P )
10 cpmidgsum.c 
 |-  C = ( N CharPlyMat R )
11 cpmidgsum.k 
 |-  K = ( C ` M )
12 cpmidgsum.h 
 |-  H = ( K .x. .1. )
13 cpmidgsumm2pm.o 
 |-  O = ( 1r ` A )
14 cpmidgsumm2pm.m 
 |-  .* = ( .s ` A )
15 cpmidgsumm2pm.t 
 |-  T = ( N matToPolyMat R )
16 cpmidpmat.g 
 |-  G = ( k e. NN0 |-> ( ( ( coe1 ` K ) ` k ) .* O ) )
17 simpl1 
 |-  ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ k e. NN0 ) -> N e. Fin )
18 crngring 
 |-  ( R e. CRing -> R e. Ring )
19 18 3ad2ant2 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> R e. Ring )
20 19 adantr 
 |-  ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ k e. NN0 ) -> R e. Ring )
21 eqid 
 |-  ( Base ` P ) = ( Base ` P )
22 10 1 2 3 21 chpmatply1 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( C ` M ) e. ( Base ` P ) )
23 11 22 eqeltrid 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> K e. ( Base ` P ) )
24 eqid 
 |-  ( coe1 ` K ) = ( coe1 ` K )
25 eqid 
 |-  ( Base ` R ) = ( Base ` R )
26 24 21 3 25 coe1fvalcl 
 |-  ( ( K e. ( Base ` P ) /\ k e. NN0 ) -> ( ( coe1 ` K ) ` k ) e. ( Base ` R ) )
27 23 26 sylan 
 |-  ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ k e. NN0 ) -> ( ( coe1 ` K ) ` k ) e. ( Base ` R ) )
28 18 anim2i 
 |-  ( ( N e. Fin /\ R e. CRing ) -> ( N e. Fin /\ R e. Ring ) )
29 1 matring 
 |-  ( ( N e. Fin /\ R e. Ring ) -> A e. Ring )
30 2 13 ringidcl 
 |-  ( A e. Ring -> O e. B )
31 28 29 30 3syl 
 |-  ( ( N e. Fin /\ R e. CRing ) -> O e. B )
32 31 3adant3 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> O e. B )
33 32 adantr 
 |-  ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ k e. NN0 ) -> O e. B )
34 25 1 2 14 matvscl 
 |-  ( ( ( N e. Fin /\ R e. Ring ) /\ ( ( ( coe1 ` K ) ` k ) e. ( Base ` R ) /\ O e. B ) ) -> ( ( ( coe1 ` K ) ` k ) .* O ) e. B )
35 17 20 27 33 34 syl22anc 
 |-  ( ( ( N e. Fin /\ R e. CRing /\ M e. B ) /\ k e. NN0 ) -> ( ( ( coe1 ` K ) ` k ) .* O ) e. B )
36 35 16 fmptd 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> G : NN0 --> B )
37 2 fvexi 
 |-  B e. _V
38 nn0ex 
 |-  NN0 e. _V
39 37 38 pm3.2i 
 |-  ( B e. _V /\ NN0 e. _V )
40 elmapg 
 |-  ( ( B e. _V /\ NN0 e. _V ) -> ( G e. ( B ^m NN0 ) <-> G : NN0 --> B ) )
41 39 40 mp1i 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> ( G e. ( B ^m NN0 ) <-> G : NN0 --> B ) )
42 36 41 mpbird 
 |-  ( ( N e. Fin /\ R e. CRing /\ M e. B ) -> G e. ( B ^m NN0 ) )