MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evls1val Structured version   Visualization version   Unicode version

Theorem evls1val 18986
Description: Value of the univariate polynomial evaluation map. (Contributed by AV, 10-Sep-2019.)
Hypotheses
Ref Expression
evls1fval.q  |-  Q  =  ( S evalSub1  R )
evls1fval.e  |-  E  =  ( 1o evalSub  S )
evls1fval.b  |-  B  =  ( Base `  S
)
evls1val.m  |-  M  =  ( 1o mPoly  ( Ss  R
) )
evls1val.k  |-  K  =  ( Base `  M
)
Assertion
Ref Expression
evls1val  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( Q `  A )  =  ( ( ( E `  R ) `  A
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
Distinct variable group:    y, B
Allowed substitution hints:    A( y)    Q( y)    R( y)    S( y)    E( y)    K( y)    M( y)

Proof of Theorem evls1val
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 evls1fval.b . . . . . . . 8  |-  B  =  ( Base `  S
)
21subrgss 18087 . . . . . . 7  |-  ( R  e.  (SubRing `  S
)  ->  R  C_  B
)
32adantl 473 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  C_  B
)
4 elpwg 3950 . . . . . . 7  |-  ( R  e.  (SubRing `  S
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
54adantl 473 . . . . . 6  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( R  e.  ~P B  <->  R  C_  B
) )
63, 5mpbird 240 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  R  e.  ~P B )
7 evls1fval.q . . . . . 6  |-  Q  =  ( S evalSub1  R )
8 evls1fval.e . . . . . 6  |-  E  =  ( 1o evalSub  S )
97, 8, 1evls1fval 18985 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  ~P B )  ->  Q  =  ( (
x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `  R
) ) )
106, 9syldan 478 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  Q  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `
 R ) ) )
1110fveq1d 5881 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( Q `  A )  =  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `
 R ) ) `
 A ) )
12113adant3 1050 . 2  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( Q `  A )  =  ( ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `
 R ) ) `
 A ) )
13 1on 7207 . . . . . 6  |-  1o  e.  On
1413a1i 11 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  1o  e.  On )
15 simp1 1030 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  S  e.  CRing
)
16 simp2 1031 . . . . 5  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  R  e.  (SubRing `  S ) )
178fveq1i 5880 . . . . . 6  |-  ( E `
 R )  =  ( ( 1o evalSub  S ) `
 R )
18 evls1val.m . . . . . 6  |-  M  =  ( 1o mPoly  ( Ss  R
) )
19 eqid 2471 . . . . . 6  |-  ( Ss  R )  =  ( Ss  R )
20 eqid 2471 . . . . . 6  |-  ( S  ^s  ( B  ^m  1o ) )  =  ( S  ^s  ( B  ^m  1o ) )
2117, 18, 19, 20, 1evlsrhm 18821 . . . . 5  |-  ( ( 1o  e.  On  /\  S  e.  CRing  /\  R  e.  (SubRing `  S )
)  ->  ( E `  R )  e.  ( M RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
2214, 15, 16, 21syl3anc 1292 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( E `  R )  e.  ( M RingHom  ( S  ^s  ( B  ^m  1o ) ) ) )
23 evls1val.k . . . . 5  |-  K  =  ( Base `  M
)
24 eqid 2471 . . . . 5  |-  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( S  ^s  ( B  ^m  1o ) ) )
2523, 24rhmf 18032 . . . 4  |-  ( ( E `  R )  e.  ( M RingHom  ( S  ^s  ( B  ^m  1o ) ) )  -> 
( E `  R
) : K --> ( Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
2622, 25syl 17 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( E `  R ) : K --> ( Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
27 simp3 1032 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  A  e.  K )
28 fvco3 5957 . . 3  |-  ( ( ( E `  R
) : K --> ( Base `  ( S  ^s  ( B  ^m  1o ) ) )  /\  A  e.  K )  ->  (
( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `
 R ) ) `
 A )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( E `  R ) `
 A ) ) )
2926, 27, 28syl2anc 673 . 2  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( (
( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )  o.  ( E `
 R ) ) `
 A )  =  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) ) 
|->  ( x  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( E `  R ) `
 A ) ) )
3026, 27ffvelrnd 6038 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( ( E `  R ) `  A )  e.  (
Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
31 ovex 6336 . . . . 5  |-  ( B  ^m  1o )  e. 
_V
3220, 1pwsbas 15463 . . . . 5  |-  ( ( S  e.  CRing  /\  ( B  ^m  1o )  e. 
_V )  ->  ( B  ^m  ( B  ^m  1o ) )  =  (
Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
3315, 31, 32sylancl 675 . . . 4  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( B  ^m  ( B  ^m  1o ) )  =  (
Base `  ( S  ^s  ( B  ^m  1o ) ) ) )
3430, 33eleqtrrd 2552 . . 3  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( ( E `  R ) `  A )  e.  ( B  ^m  ( B  ^m  1o ) ) )
35 coeq1 4997 . . . 4  |-  ( x  =  ( ( E `
 R ) `  A )  ->  (
x  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) )  =  ( ( ( E `
 R ) `  A )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
36 eqid 2471 . . . 4  |-  ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )  =  ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
37 fvex 5889 . . . . 5  |-  ( ( E `  R ) `
 A )  e. 
_V
38 fvex 5889 . . . . . . 7  |-  ( Base `  S )  e.  _V
391, 38eqeltri 2545 . . . . . 6  |-  B  e. 
_V
4039mptex 6152 . . . . 5  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) )  e.  _V
4137, 40coex 6764 . . . 4  |-  ( ( ( E `  R
) `  A )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  e.  _V
4235, 36, 41fvmpt 5963 . . 3  |-  ( ( ( E `  R
) `  A )  e.  ( B  ^m  ( B  ^m  1o ) )  ->  ( ( x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) ) `  (
( E `  R
) `  A )
)  =  ( ( ( E `  R
) `  A )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
4334, 42syl 17 . 2  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( (
x  e.  ( B  ^m  ( B  ^m  1o ) )  |->  ( x  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) ) ) `  ( ( E `  R ) `  A
) )  =  ( ( ( E `  R ) `  A
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
4412, 29, 433eqtrd 2509 1  |-  ( ( S  e.  CRing  /\  R  e.  (SubRing `  S )  /\  A  e.  K
)  ->  ( Q `  A )  =  ( ( ( E `  R ) `  A
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031    C_ wss 3390   ~Pcpw 3942   {csn 3959    |-> cmpt 4454    X. cxp 4837    o. ccom 4843   Oncon0 5430   -->wf 5585   ` cfv 5589  (class class class)co 6308   1oc1o 7193    ^m cmap 7490   Basecbs 15199   ↾s cress 15200    ^s cpws 15423   CRingccrg 17859   RingHom crh 18018  SubRingcsubrg 18082   mPoly cmpl 18654   evalSub ces 18804   evalSub1 ces1 18979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-rnghom 18021  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-assa 18613  df-asp 18614  df-ascl 18615  df-psr 18657  df-mvr 18658  df-mpl 18659  df-evls 18806  df-evls1 18981
This theorem is referenced by:  evls1var  19003
  Copyright terms: Public domain W3C validator