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Theorem mpfpf1 17785
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 2443 . . . . . . 7  |-  ( 1o eval  R )  =  ( 1o eval  R )
3 pf1f.b . . . . . . 7  |-  B  =  ( Base `  R
)
42, 3evlval 17610 . . . . . 6  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
54rneqi 5066 . . . . 5  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 B )
61, 5eqtri 2463 . . . 4  |-  E  =  ran  ( ( 1o evalSub  R ) `  B
)
76mpfrcl 17604 . . 3  |-  ( F  e.  E  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  B  e.  (SubRing `  R ) ) )
87simp2d 1001 . 2  |-  ( F  e.  E  ->  R  e.  CRing )
9 id 22 . . . 4  |-  ( F  e.  E  ->  F  e.  E )
109, 1syl6eleq 2533 . . 3  |-  ( F  e.  E  ->  F  e.  ran  ( 1o eval  R
) )
11 1on 6927 . . . . 5  |-  1o  e.  On
12 eqid 2443 . . . . . 6  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
13 eqid 2443 . . . . . 6  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
142, 3, 12, 13evlrhm 17611 . . . . 5  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
1511, 8, 14sylancr 663 . . . 4  |-  ( F  e.  E  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
16 eqid 2443 . . . . . 6  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
17 eqid 2443 . . . . . 6  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
18 eqid 2443 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
1916, 17, 18ply1bas 17651 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
20 eqid 2443 . . . . 5  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
2119, 20rhmf 16816 . . . 4  |-  ( ( 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 ) ) ) )
22 ffn 5559 . . . 4  |-  ( ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
23 fvelrnb 5739 . . . 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 ) )
2415, 21, 22, 234syl 21 . . 3  |-  ( F  e.  E  ->  ( F  e.  ran  ( 1o eval  R )  <->  E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R ) `
 x )  =  F ) )
2510, 24mpbid 210 . 2  |-  ( F  e.  E  ->  E. x  e.  ( Base `  (Poly1 `  R ) ) ( ( 1o eval  R ) `
 x )  =  F )
26 eqid 2443 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
2726, 2, 3, 12, 19evl1val 17763 . . . . 5  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  =  ( ( ( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) ) )
28 eqid 2443 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
2926, 16, 28, 3evl1rhm 17766 . . . . . . . 8  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
30 eqid 2443 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
3118, 30rhmf 16816 . . . . . . . 8  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
32 ffn 5559 . . . . . . . 8  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
3329, 31, 323syl 20 . . . . . . 7  |-  ( R  e.  CRing  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
34 fnfvelrn 5840 . . . . . . 7  |-  ( ( (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R ) `  x
)  e.  ran  (eval1 `  R ) )
3533, 34sylan 471 . . . . . 6  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  e.  ran  (eval1 `  R ) )
36 pf1rcl.q . . . . . 6  |-  Q  =  ran  (eval1 `  R )
3735, 36syl6eleqr 2534 . . . . 5  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  e.  Q )
3827, 37eqeltrrd 2518 . . . 4  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q )
39 coeq1 4997 . . . . 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 } ) ) ) )
4039eleq1d 2509 . . . 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 ) )
4138, 40syl5ibcom 220 . . 3  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  x )  =  F  ->  ( F  o.  ( y  e.  B  |->  ( 1o  X.  {
y } ) ) )  e.  Q ) )
4241rexlimdva 2841 . 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
) )
438, 25, 42sylc 60 1  |-  ( F  e.  E  ->  ( F  o.  ( y  e.  B  |->  ( 1o 
X.  { y } ) ) )  e.  Q )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   {csn 3877    e. cmpt 4350   Oncon0 4719    X. cxp 4838   ran crn 4841    o. ccom 4844    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   1oc1o 6913    ^m cmap 7214   Basecbs 14174    ^s cpws 14385   CRingccrg 16646   RingHom crh 16804  SubRingcsubrg 16861   mPoly cmpl 17420   evalSub ces 17586   eval cevl 17587  PwSer1cps1 17631  Poly1cpl1 17633  eval1ce1 17749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-ofr 6321  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-map 7216  df-pm 7217  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-sup 7691  df-oi 7724  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-fz 11438  df-fzo 11549  df-seq 11807  df-hash 12104  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-hom 14262  df-cco 14263  df-0g 14380  df-gsum 14381  df-prds 14386  df-pws 14388  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-mhm 15464  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-subg 15678  df-ghm 15745  df-cntz 15835  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-srg 16608  df-rng 16647  df-cring 16648  df-rnghom 16806  df-subrg 16863  df-lmod 16950  df-lss 17014  df-lsp 17053  df-assa 17384  df-asp 17385  df-ascl 17386  df-psr 17423  df-mvr 17424  df-mpl 17425  df-opsr 17427  df-evls 17588  df-evl 17589  df-psr1 17636  df-ply1 17638  df-evl1 17751
This theorem is referenced by:  pf1ind  17789
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