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Theorem evl1sca 19911
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 15637 . . . . . 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 2412 . . . . . 6  |-  ( Base `  P )  =  (
Base `  P )
73, 4, 5, 6ply1sclf 16640 . . . . 5  |-  ( R  e.  Ring  ->  A : B
--> ( Base `  P
) )
82, 7syl 16 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A : B --> ( Base `  P
) )
9 ffvelrn 5835 . . . 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 2412 . . . 4  |-  ( 1o eval  R )  =  ( 1o eval  R )
13 eqid 2412 . . . 4  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
14 eqid 2412 . . . . 5  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
153, 14, 6ply1bas 16556 . . . 4  |-  ( Base `  P )  =  (
Base `  ( 1o mPoly  R ) )
1611, 12, 5, 13, 15evl1val 19909 . . 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 13487 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( Rs  B
)  =  R )
1918adantr 452 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( Rs  B )  =  R )
2019oveq2d 6064 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( 1o mPoly  ( Rs  B ) )  =  ( 1o mPoly  R )
)
2120fveq2d 5699 . . . . . . 7  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  R ) ) )
223, 4ply1ascl 16614 . . . . . . 7  |-  A  =  (algSc `  ( 1o mPoly  R ) )
2321, 22syl6reqr 2463 . . . . . 6  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  A  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) ) )
2423fveq1d 5697 . . . . 5  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  ( A `  X )  =  ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) )
2524fveq2d 5699 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( 1o eval  R ) `
 ( (algSc `  ( 1o mPoly  ( Rs  B
) ) ) `  X ) ) )
2612, 5evlval 19906 . . . . 5  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
27 eqid 2412 . . . . 5  |-  ( 1o mPoly 
( Rs  B ) )  =  ( 1o mPoly  ( Rs  B
) )
28 eqid 2412 . . . . 5  |-  ( Rs  B )  =  ( Rs  B )
29 eqid 2412 . . . . 5  |-  (algSc `  ( 1o mPoly  ( Rs  B
) ) )  =  (algSc `  ( 1o mPoly  ( Rs  B ) ) )
30 1on 6698 . . . . . 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 15833 . . . . . 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 19904 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( (algSc `  ( 1o mPoly  ( Rs  B ) ) ) `
 X ) )  =  ( ( B  ^m  1o )  X. 
{ X } ) )
3725, 36eqtrd 2444 . . 3  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
( 1o eval  R ) `  ( A `  X
) )  =  ( ( B  ^m  1o )  X.  { X }
) )
3837coeq1d 5001 . 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 6703 . . . . . . 7  |-  1o  =  { (/) }
40 fvex 5709 . . . . . . . 8  |-  ( Base `  R )  e.  _V
415, 40eqeltri 2482 . . . . . . 7  |-  B  e. 
_V
42 0ex 4307 . . . . . . 7  |-  (/)  e.  _V
43 eqid 2412 . . . . . . 7  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  { y } ) )
4439, 41, 42, 43mapsnf1o3 7029 . . . . . 6  |-  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) : B -1-1-onto-> ( B  ^m  1o )
45 f1of 5641 . . . . . 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 5857 . . . . 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 2413 . . . 4  |-  ( ( R  e.  CRing  /\  X  e.  B )  ->  (
y  e.  B  |->  ( 1o  X.  { y } ) )  =  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )
50 fconstmpt 4888 . . . . 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 2413 . . . 4  |-  ( x  =  ( 1o  X.  { y } )  ->  X  =  X )
5348, 49, 51, 52fmptcof 5869 . . 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 4888 . . 3  |-  ( B  X.  { X }
)  =  ( y  e.  B  |->  X )
5553, 54syl6eqr 2462 . 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 2448 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 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924   (/)c0 3596   {csn 3782    e. cmpt 4234   Oncon0 4549    X. cxp 4843    o. ccom 4849   -->wf 5417   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6048   1oc1o 6684    ^m cmap 6985   Basecbs 13432   ↾s cress 13433   Ringcrg 15623   CRingccrg 15624  SubRingcsubrg 15827  algSccascl 16334   mPoly cmpl 16371   eval cevl 16373  PwSer1cps1 16532  Poly1cpl1 16534  eval1ce1 16536
This theorem is referenced by:  evl1scad  19912  pf1const  19927  pf1ind  19936  ply1rem  20047  fta1g  20051  fta1blem  20052  plypf1  20092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-ofr 6273  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-ixp 7031  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-fz 11008  df-fzo 11099  df-seq 11287  df-hash 11582  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-tset 13511  df-ple 13512  df-ds 13514  df-hom 13516  df-cco 13517  df-prds 13634  df-pws 13636  df-0g 13690  df-gsum 13691  df-mre 13774  df-mrc 13775  df-acs 13777  df-mnd 14653  df-mhm 14701  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-mulg 14778  df-subg 14904  df-ghm 14967  df-cntz 15079  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-cring 15627  df-ur 15628  df-rnghom 15782  df-subrg 15829  df-lmod 15915  df-lss 15972  df-lsp 16011  df-assa 16335  df-asp 16336  df-ascl 16337  df-psr 16380  df-mvr 16381  df-mpl 16382  df-evls 16383  df-evl 16384  df-opsr 16388  df-psr1 16539  df-ply1 16541  df-evl1 16543
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