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Theorem mpfpf1 17754
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 2441 . . . . . . 7  |-  ( 1o eval  R )  =  ( 1o eval  R )
3 pf1f.b . . . . . . 7  |-  B  =  ( Base `  R
)
42, 3evlval 17586 . . . . . 6  |-  ( 1o eval  R )  =  ( ( 1o evalSub  R ) `  B )
54rneqi 5062 . . . . 5  |-  ran  ( 1o eval  R )  =  ran  ( ( 1o evalSub  R ) `
 B )
61, 5eqtri 2461 . . . 4  |-  E  =  ran  ( ( 1o evalSub  R ) `  B
)
76mpfrcl 17580 . . 3  |-  ( F  e.  E  ->  ( 1o  e.  _V  /\  R  e.  CRing  /\  B  e.  (SubRing `  R ) ) )
87simp2d 996 . 2  |-  ( F  e.  E  ->  R  e.  CRing )
9 id 22 . . . 4  |-  ( F  e.  E  ->  F  e.  E )
109, 1syl6eleq 2531 . . 3  |-  ( F  e.  E  ->  F  e.  ran  ( 1o eval  R
) )
11 1on 6923 . . . . 5  |-  1o  e.  On
12 eqid 2441 . . . . . 6  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
13 eqid 2441 . . . . . 6  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
142, 3, 12, 13evlrhm 17587 . . . . 5  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
1511, 8, 14sylancr 658 . . . 4  |-  ( F  e.  E  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
16 eqid 2441 . . . . . 6  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
17 eqid 2441 . . . . . 6  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
18 eqid 2441 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
1916, 17, 18ply1bas 17627 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
20 eqid 2441 . . . . 5  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
2119, 20rhmf 16804 . . . 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 5556 . . . 4  |-  ( ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
23 fvelrnb 5736 . . . 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 2441 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
2726, 2, 3, 12, 19evl1val 17733 . . . . 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 2441 . . . . . . . . 9  |-  ( R  ^s  B )  =  ( R  ^s  B )
2926, 16, 28, 3evl1rhm 17736 . . . . . . . 8  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
30 eqid 2441 . . . . . . . . 9  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
3118, 30rhmf 16804 . . . . . . . 8  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
32 ffn 5556 . . . . . . . 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 5837 . . . . . . 7  |-  ( ( (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R ) `  x
)  e.  ran  (eval1 `  R ) )
3533, 34sylan 468 . . . . . 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 2532 . . . . 5  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  x )  e.  Q )
3827, 37eqeltrrd 2516 . . . 4  |-  ( ( R  e.  CRing  /\  x  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  x )  o.  (
y  e.  B  |->  ( 1o  X.  { y } ) ) )  e.  Q )
39 coeq1 4993 . . . . 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 2507 . . . 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 2839 . 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 1364    e. wcel 1761   E.wrex 2714   _Vcvv 2970   {csn 3874    e. cmpt 4347   Oncon0 4715    X. cxp 4834   ran crn 4837    o. ccom 4840    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   1oc1o 6909    ^m cmap 7210   Basecbs 14170    ^s cpws 14381   CRingccrg 16636   RingHom crh 16794  SubRingcsubrg 16841   mPoly cmpl 17398   evalSub ces 17562   eval cevl 17563  PwSer1cps1 17607  Poly1cpl1 17609  eval1ce1 17722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-ofr 6320  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-sup 7687  df-oi 7720  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-fz 11434  df-fzo 11545  df-seq 11803  df-hash 12100  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-hom 14258  df-cco 14259  df-0g 14376  df-gsum 14377  df-prds 14382  df-pws 14384  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-sbg 15540  df-mulg 15541  df-subg 15671  df-ghm 15738  df-cntz 15828  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-rnghom 16796  df-subrg 16843  df-lmod 16930  df-lss 16992  df-lsp 17031  df-assa 17362  df-asp 17363  df-ascl 17364  df-psr 17401  df-mvr 17402  df-mpl 17403  df-opsr 17405  df-evls 17564  df-evl 17565  df-psr1 17612  df-ply1 17614  df-evl1 17724
This theorem is referenced by:  pf1ind  17758
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