MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  evl1sca Unicode version

Theorem evl1sca 19628
Description: Polynomial evaluation maps scalars to constant functions. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
evl1sca.o  |-  O  =  (eval1 `  R )
evl1sca.p  |-  P  =  (Poly1 `  R )
evl1sca.b  |-  B  =  ( Base `  R
)
evl1sca.a  |-  A  =  (algSc `  P )
Assertion
Ref Expression
evl1sca  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( O `  ( A `  X ) )  =  ( B  X.  { X } ) )

Proof of Theorem evl1sca
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 crngrng 15561 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 451 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  R  e.  Ring )
3 evl1sca.p . . . . . 6  |-  P  =  (Poly1 `  R )
4 evl1sca.a . . . . . 6  |-  A  =  (algSc `  P )
5 evl1sca.b . . . . . 6  |-  B  =  ( Base `  R
)
6 eqid 2366 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
73, 4, 5, 6ply1sclf 16571 . . . . 5  |-  ( R  e.  Ring  ->  A : B
--> ( Base `  P
) )
82, 7syl 15 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A : B --> ( Base `  P
) )
9 ffvelrn 5770 . . . 4  |-  ( ( A : B --> ( Base `  P )  /\  X  e.  B )  ->  ( A `  X )  e.  ( Base `  P
) )
108, 9sylancom 648 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( A `  X )  e.  ( Base `  P
) )
11 evl1sca.o . . . 4  |-  O  =  (eval1 `  R )
12 eqid 2366 . . . 4  |-  ( 1o eval  R )  =  ( 1o eval  R )
13 eqid 2366 . . . 4  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
14 eqid 2366 . . . . 5  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
153, 14, 6ply1bas 16484 . . . 4  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
1611, 12, 5, 13, 15evl1val 19626 . . 3  |-  ( ( R  e.  CRing  /\  ( A `  X )  e.  ( Base `  P
) )  ->  ( O `  ( A `  X ) )  =  ( ( ( 1o eval  R ) `  ( A `  X )
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
1710, 16syldan 456 . 2  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( O `  ( A `  X ) )  =  ( ( ( 1o eval  R ) `  ( A `  X )
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
185ressid 13411 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( Rs  B
)  =  R )
1918adantr 451 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( Rs  B )  =  R )
2019oveq2d 5997 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( 1o mPoly  ( Rs  B ) )  =  ( 1o mPoly  R )
)
2120fveq2d 5636 . . . . . . 7  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  R ) ) )
223, 4ply1ascl 16545 . . . . . . 7  |-  A  =  (algSc `  ( 1o mPoly  R ) )
2321, 22syl6reqr 2417 . . . . . 6  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) ) )
2423fveq1d 5634 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( A `  X )  =  ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) )
2524fveq2d 5636 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( 1o eval  R ) `
 ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) ) )
2612, 5evlval 19623 . . . . 5  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
27 eqid 2366 . . . . 5  |-  ( 1o mPoly 
( Rs  B ) )  =  ( 1o mPoly  ( Rs  B
) )
28 eqid 2366 . . . . 5  |-  ( Rs  B )  =  ( Rs  B )
29 eqid 2366 . . . . 5  |-  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) )
30 1on 6628 . . . . . 6  |-  1o  e.  On
3130a1i 10 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  1o  e.  On )
32 simpl 443 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  R  e.  CRing )
335subrgid 15757 . . . . . 6  |-  ( R  e.  Ring  ->  B  e.  (SubRing `  R )
)
342, 33syl 15 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  B  e.  (SubRing `  R )
)
35 simpr 447 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  X  e.  B )
3626, 27, 28, 5, 29, 31, 32, 34, 35evlssca 19621 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( (algSc `  ( 1o mPoly  ( Rs  B ) ) ) `
 X ) )  =  ( ( B  ^m  1o )  X. 
{ X } ) )
3725, 36eqtrd 2398 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( B  ^m  1o )  X.  { X }
) )
3837coeq1d 4948 . 2  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( ( 1o eval  R
) `  ( A `  X ) )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  =  ( ( ( B  ^m  1o )  X.  { X }
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
39 df1o2 6633 . . . . . . 7  |-  1o  =  { (/) }
40 fvex 5646 . . . . . . . 8  |-  ( Base `  R )  e.  _V
415, 40eqeltri 2436 . . . . . . 7  |-  B  e. 
_V
42 0ex 4252 . . . . . . 7  |-  (/)  e.  _V
43 eqid 2366 . . . . . . 7  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  { y } ) )
4439, 41, 42, 43mapsnf1o3 6959 . . . . . 6  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) : B -1-1-onto-> ( B  ^m  1o )
45 f1of 5578 . . . . . 6  |-  ( ( y  e.  B  |->  ( 1o  X.  { y } ) ) : B -1-1-onto-> ( B  ^m  1o )  ->  ( y  e.  B  |->  ( 1o  X.  { y } ) ) : B --> ( B  ^m  1o ) )
4644, 45mp1i 11 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) ) : B --> ( B  ^m  1o ) )
4743fmpt 5792 . . . . 5  |-  ( A. y  e.  B  ( 1o  X.  { y } )  e.  ( B  ^m  1o )  <->  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) : B --> ( B  ^m  1o ) )
4846, 47sylibr 203 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A. y  e.  B  ( 1o  X.  { y } )  e.  ( B  ^m  1o ) )
49 eqidd 2367 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )
50 fconstmpt 4835 . . . . 5  |-  ( ( B  ^m  1o )  X.  { X }
)  =  ( x  e.  ( B  ^m  1o )  |->  X )
5150a1i 10 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( B  ^m  1o )  X.  { X }
)  =  ( x  e.  ( B  ^m  1o )  |->  X ) )
52 eqidd 2367 . . . 4  |-  ( x  =  ( 1o  X.  { y } )  ->  X  =  X )
5348, 49, 51, 52fmptcof 5803 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( ( B  ^m  1o )  X.  { X } )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( y  e.  B  |->  X ) )
54 fconstmpt 4835 . . 3  |-  ( B  X.  { X }
)  =  ( y  e.  B  |->  X )
5553, 54syl6eqr 2416 . 2  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( ( B  ^m  1o )  X.  { X } )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( B  X.  { X } ) )
5617, 38, 553eqtrd 2402 1  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( O `  ( A `  X ) )  =  ( B  X.  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873   (/)c0 3543   {csn 3729    e. cmpt 4179   Oncon0 4495    X. cxp 4790    o. ccom 4796   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   1oc1o 6614    ^m cmap 6915   Basecbs 13356   ↾s cress 13357   Ringcrg 15547   CRingccrg 15548  SubRingcsubrg 15751  algSccascl 16262   mPoly cmpl 16299   eval cevl 16301  PwSer1cps1 16460  Poly1cpl1 16462  eval1ce1 16464
This theorem is referenced by:  evl1scad  19629  pf1const  19644  pf1ind  19653  ply1rem  19764  fta1g  19768  fta1blem  19769  plypf1  19809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-ofr 6206  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-oi 7372  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-fz 10936  df-fzo 11026  df-seq 11211  df-hash 11506  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-hom 13440  df-cco 13441  df-prds 13558  df-pws 13560  df-0g 13614  df-gsum 13615  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-mhm 14625  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-mulg 14702  df-subg 14828  df-ghm 14891  df-cntz 15003  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-cring 15551  df-ur 15552  df-rnghom 15706  df-subrg 15753  df-lmod 15839  df-lss 15900  df-lsp 15939  df-assa 16263  df-asp 16264  df-ascl 16265  df-psr 16308  df-mvr 16309  df-mpl 16310  df-evls 16311  df-evl 16312  df-opsr 16316  df-psr1 16467  df-ply1 16469  df-evl1 16471
  Copyright terms: Public domain W3C validator