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Theorem evl1sca 19938
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 15662 . . . . . 6  |-  ( R  e.  CRing  ->  R  e.  Ring )
21adantr 452 . . . . 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 2435 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
73, 4, 5, 6ply1sclf 16665 . . . . 5  |-  ( R  e.  Ring  ->  A : B
--> ( Base `  P
) )
82, 7syl 16 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A : B --> ( Base `  P
) )
9 ffvelrn 5859 . . . 4  |-  ( ( A : B --> ( Base `  P )  /\  X  e.  B )  ->  ( A `  X )  e.  ( Base `  P
) )
108, 9sylancom 649 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( A `  X )  e.  ( Base `  P
) )
11 evl1sca.o . . . 4  |-  O  =  (eval1 `  R )
12 eqid 2435 . . . 4  |-  ( 1o eval  R )  =  ( 1o eval  R )
13 eqid 2435 . . . 4  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
14 eqid 2435 . . . . 5  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
153, 14, 6ply1bas 16581 . . . 4  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
1611, 12, 5, 13, 15evl1val 19936 . . 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 457 . 2  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( O `  ( A `  X ) )  =  ( ( ( 1o eval  R ) `  ( A `  X )
)  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) ) )
185ressid 13512 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( Rs  B
)  =  R )
1918adantr 452 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( Rs  B )  =  R )
2019oveq2d 6088 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( 1o mPoly  ( Rs  B ) )  =  ( 1o mPoly  R )
)
2120fveq2d 5723 . . . . . . 7  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  R ) ) )
223, 4ply1ascl 16639 . . . . . . 7  |-  A  =  (algSc `  ( 1o mPoly  R ) )
2321, 22syl6reqr 2486 . . . . . 6  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) ) )
2423fveq1d 5721 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( A `  X )  =  ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) )
2524fveq2d 5723 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( 1o eval  R ) `
 ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) ) )
2612, 5evlval 19933 . . . . 5  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
27 eqid 2435 . . . . 5  |-  ( 1o mPoly 
( Rs  B ) )  =  ( 1o mPoly  ( Rs  B
) )
28 eqid 2435 . . . . 5  |-  ( Rs  B )  =  ( Rs  B )
29 eqid 2435 . . . . 5  |-  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) )
30 1on 6722 . . . . . 6  |-  1o  e.  On
3130a1i 11 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  1o  e.  On )
32 simpl 444 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  R  e.  CRing )
335subrgid 15858 . . . . . 6  |-  ( R  e.  Ring  ->  B  e.  (SubRing `  R )
)
342, 33syl 16 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  B  e.  (SubRing `  R )
)
35 simpr 448 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  X  e.  B )
3626, 27, 28, 5, 29, 31, 32, 34, 35evlssca 19931 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( (algSc `  ( 1o mPoly  ( Rs  B ) ) ) `
 X ) )  =  ( ( B  ^m  1o )  X. 
{ X } ) )
3725, 36eqtrd 2467 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( B  ^m  1o )  X.  { X }
) )
3837coeq1d 5025 . 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 6727 . . . . . . 7  |-  1o  =  { (/) }
40 fvex 5733 . . . . . . . 8  |-  ( Base `  R )  e.  _V
415, 40eqeltri 2505 . . . . . . 7  |-  B  e. 
_V
42 0ex 4331 . . . . . . 7  |-  (/)  e.  _V
43 eqid 2435 . . . . . . 7  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  { y } ) )
4439, 41, 42, 43mapsnf1o3 7053 . . . . . 6  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) : B -1-1-onto-> ( B  ^m  1o )
45 f1of 5665 . . . . . 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 12 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) ) : B --> ( B  ^m  1o ) )
4743fmpt 5881 . . . . 5  |-  ( A. y  e.  B  ( 1o  X.  { y } )  e.  ( B  ^m  1o )  <->  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) : B --> ( B  ^m  1o ) )
4846, 47sylibr 204 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A. y  e.  B  ( 1o  X.  { y } )  e.  ( B  ^m  1o ) )
49 eqidd 2436 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )
50 fconstmpt 4912 . . . . 5  |-  ( ( B  ^m  1o )  X.  { X }
)  =  ( x  e.  ( B  ^m  1o )  |->  X )
5150a1i 11 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( B  ^m  1o )  X.  { X }
)  =  ( x  e.  ( B  ^m  1o )  |->  X ) )
52 eqidd 2436 . . . 4  |-  ( x  =  ( 1o  X.  { y } )  ->  X  =  X )
5348, 49, 51, 52fmptcof 5893 . . 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 4912 . . 3  |-  ( B  X.  { X }
)  =  ( y  e.  B  |->  X )
5553, 54syl6eqr 2485 . 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 2471 1  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( O `  ( A `  X ) )  =  ( B  X.  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   (/)c0 3620   {csn 3806    e. cmpt 4258   Oncon0 4573    X. cxp 4867    o. ccom 4873   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072   1oc1o 6708    ^m cmap 7009   Basecbs 13457   ↾s cress 13458   Ringcrg 15648   CRingccrg 15649  SubRingcsubrg 15852  algSccascl 16359   mPoly cmpl 16396   eval cevl 16398  PwSer1cps1 16557  Poly1cpl1 16559  eval1ce1 16561
This theorem is referenced by:  evl1scad  19939  pf1const  19954  pf1ind  19963  ply1rem  20074  fta1g  20078  fta1blem  20079  plypf1  20119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-ofr 6297  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-fz 11033  df-fzo 11124  df-seq 11312  df-hash 11607  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-hom 13541  df-cco 13542  df-prds 13659  df-pws 13661  df-0g 13715  df-gsum 13716  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-mhm 14726  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-mulg 14803  df-subg 14929  df-ghm 14992  df-cntz 15104  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-cring 15652  df-ur 15653  df-rnghom 15807  df-subrg 15854  df-lmod 15940  df-lss 15997  df-lsp 16036  df-assa 16360  df-asp 16361  df-ascl 16362  df-psr 16405  df-mvr 16406  df-mpl 16407  df-evls 16408  df-evl 16409  df-opsr 16413  df-psr1 16564  df-ply1 16566  df-evl1 16568
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