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Theorem mpfpf1 19924
Description: Convert a multivariate polynomial function to univariate. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
pf1rcl.q  |-  Q  =  ran  (eval1 `  R )
pf1f.b  |-  B  =  ( Base `  R
)
mpfpf1.q  |-  E  =  ran  ( 1o eval  R
)
Assertion
Ref Expression
mpfpf1  |-  ( F  e.  E  ->  ( F  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) )  e.  Q )
Distinct variable groups:    y, B    y, E    y, F    y, R
Allowed substitution hint:    Q( y)

Proof of Theorem mpfpf1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mpfpf1.q . . . . 5  |-  E  =  ran  ( 1o eval  R
)
2 eqid 2404 . . . . . . 7  |-  ( 1o eval  R )  =  ( 1o eval  R )
3 pf1f.b . . . . . . 7  |-  B  =  ( Base `  R
)
42, 3evlval 19898 . . . . . 6  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
54rneqi 5055 . . . . 5  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 B )
61, 5eqtri 2424 . . . 4  |-  E  =  ran  ( ( 1o evalSub  R ) `  B
)
76mpfrcl 19892 . . 3  |-  ( F  e.  E  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  B  e.  (SubRing `  R ) ) )
87simp2d 970 . 2  |-  ( F  e.  E  ->  R  e.  CRing )
9 id 20 . . . 4  |-  ( F  e.  E  ->  F  e.  E )
109, 1syl6eleq 2494 . . 3  |-  ( F  e.  E  ->  F  e.  ran  ( 1o eval  R
) )
11 1on 6690 . . . . . 6  |-  1o  e.  On
12 eqid 2404 . . . . . . 7  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
13 eqid 2404 . . . . . . 7  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
142, 3, 12, 13evlrhm 19899 . . . . . 6  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
1511, 8, 14sylancr 645 . . . . 5  |-  ( F  e.  E  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
16 eqid 2404 . . . . . . 7  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
17 eqid 2404 . . . . . . 7  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
18 eqid 2404 . . . . . . 7  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
1916, 17, 18ply1bas 16548 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
20 eqid 2404 . . . . . 6  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
2119, 20rhmf 15782 . . . . 5  |-  ( ( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
2215, 21syl 16 . . . 4  |-  ( F  e.  E  ->  ( 1o eval  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
23 ffn 5550 . . . 4  |-  ( ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
24 fvelrnb 5733 . . . 4  |-  ( ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) )  -> 
( F  e.  ran  ( 1o eval  R )  <->  E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R
) `  x )  =  F ) )
2522, 23, 243syl 19 . . 3  |-  ( F  e.  E  ->  ( F  e.  ran  ( 1o eval  R )  <->  E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R ) `
 x )  =  F ) )
2610, 25mpbid 202 . 2  |-  ( F  e.  E  ->  E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R ) `
 x )  =  F )
27 eqid 2404 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
2827, 2, 3, 12, 19evl1val 19901 . . . . 5  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  =  ( ( ( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
29 eqid 2404 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
3027, 16, 29, 3evl1rhm 19902 . . . . . . . 8  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
31 eqid 2404 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
3218, 31rhmf 15782 . . . . . . . 8  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
33 ffn 5550 . . . . . . . 8  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
3430, 32, 333syl 19 . . . . . . 7  |-  ( R  e.  CRing  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
35 fnfvelrn 5826 . . . . . . 7  |-  ( ( (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R ) `  x
)  e.  ran  (eval1 `  R ) )
3634, 35sylan 458 . . . . . 6  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  e.  ran  (eval1 `  R ) )
37 pf1rcl.q . . . . . 6  |-  Q  =  ran  (eval1 `  R )
3836, 37syl6eleqr 2495 . . . . 5  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  e.  Q )
3928, 38eqeltrrd 2479 . . . 4  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q )
40 coeq1 4989 . . . . 5  |-  ( ( ( 1o eval  R ) `
 x )  =  F  ->  ( (
( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  =  ( F  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) ) )
4140eleq1d 2470 . . . 4  |-  ( ( ( 1o eval  R ) `
 x )  =  F  ->  ( (
( ( 1o eval  R
) `  x )  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  e.  Q  <->  ( F  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  e.  Q ) )
4239, 41syl5ibcom 212 . . 3  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  x )  =  F  ->  ( F  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  e.  Q ) )
4342rexlimdva 2790 . 2  |-  ( R  e.  CRing  ->  ( E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R
) `  x )  =  F  ->  ( F  o.  ( y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q
) )
448, 26, 43sylc 58 1  |-  ( F  e.  E  ->  ( F  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) )  e.  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916   {csn 3774    e. cmpt 4226   Oncon0 4541    X. cxp 4835   ran crn 4838    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ^m cmap 6977   Basecbs 13424    ^s cpws 13625   CRingccrg 15616   RingHom crh 15772  SubRingcsubrg 15819   mPoly cmpl 16363   evalSub ces 16364   eval cevl 16365  PwSer1cps1 16524  Poly1cpl1 16526  eval1ce1 16528
This theorem is referenced by:  pf1ind  19928
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-assa 16327  df-asp 16328  df-ascl 16329  df-psr 16372  df-mvr 16373  df-mpl 16374  df-evls 16375  df-evl 16376  df-opsr 16380  df-psr1 16531  df-ply1 16533  df-evl1 16535
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