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Theorem pf1mpf 19925
Description: Convert a univariate polynomial function to multivariate. (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
pf1mpf  |-  ( F  e.  Q  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E )
Distinct variable groups:    x, B    x, F    x, Q    x, R
Allowed substitution hint:    E( x)

Proof of Theorem pf1mpf
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pf1rcl.q . . 3  |-  Q  =  ran  (eval1 `  R )
21pf1rcl 19922 . 2  |-  ( F  e.  Q  ->  R  e.  CRing )
3 id 20 . . . 4  |-  ( F  e.  Q  ->  F  e.  Q )
43, 1syl6eleq 2494 . . 3  |-  ( F  e.  Q  ->  F  e.  ran  (eval1 `  R ) )
5 eqid 2404 . . . . . . . 8  |-  (eval1 `  R
)  =  (eval1 `  R
)
6 eqid 2404 . . . . . . . 8  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
7 eqid 2404 . . . . . . . 8  |-  ( R  ^s  B )  =  ( R  ^s  B )
8 pf1f.b . . . . . . . 8  |-  B  =  ( Base `  R
)
95, 6, 7, 8evl1rhm 19902 . . . . . . 7  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
102, 9syl 16 . . . . . 6  |-  ( F  e.  Q  ->  (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
11 eqid 2404 . . . . . . 7  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
12 eqid 2404 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
1311, 12rhmf 15782 . . . . . 6  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
1410, 13syl 16 . . . . 5  |-  ( F  e.  Q  ->  (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) ) )
15 ffn 5550 . . . . 5  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
1614, 15syl 16 . . . 4  |-  ( F  e.  Q  ->  (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) ) )
17 fvelrnb 5733 . . . 4  |-  ( (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) )  ->  ( F  e. 
ran  (eval1 `  R )  <->  E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F ) )
1816, 17syl 16 . . 3  |-  ( F  e.  Q  ->  ( F  e.  ran  (eval1 `  R
)  <->  E. y  e.  (
Base `  (Poly1 `  R
) ) ( (eval1 `  R ) `  y
)  =  F ) )
194, 18mpbid 202 . 2  |-  ( F  e.  Q  ->  E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F )
20 eqid 2404 . . . . . . . 8  |-  ( 1o eval  R )  =  ( 1o eval  R )
21 eqid 2404 . . . . . . . 8  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
22 eqid 2404 . . . . . . . . 9  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
236, 22, 11ply1bas 16548 . . . . . . . 8  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
245, 20, 8, 21, 23evl1val 19901 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  y )  =  ( ( ( 1o eval  R ) `  y )  o.  (
z  e.  B  |->  ( 1o  X.  { z } ) ) ) )
2524coeq1d 4993 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( ( ( ( 1o eval  R ) `
 y )  o.  ( z  e.  B  |->  ( 1o  X.  {
z } ) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) ) )
26 coass 5347 . . . . . . 7  |-  ( ( ( ( 1o eval  R
) `  y )  o.  ( z  e.  B  |->  ( 1o  X.  {
z } ) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( ( ( 1o eval  R ) `  y )  o.  (
( z  e.  B  |->  ( 1o  X.  {
z } ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
27 df1o2 6695 . . . . . . . . . . 11  |-  1o  =  { (/) }
28 fvex 5701 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
298, 28eqeltri 2474 . . . . . . . . . . 11  |-  B  e. 
_V
30 0ex 4299 . . . . . . . . . . 11  |-  (/)  e.  _V
31 eqid 2404 . . . . . . . . . . 11  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) )  =  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )
3227, 29, 30, 31mapsncnv 7019 . . . . . . . . . 10  |-  `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  =  ( z  e.  B  |->  ( 1o  X.  { z } ) )
3332coeq1i 4991 . . . . . . . . 9  |-  ( `' ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  =  ( ( z  e.  B  |->  ( 1o  X.  { z } ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )
3427, 29, 30, 31mapsnf1o2 7020 . . . . . . . . . 10  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) : ( B  ^m  1o )
-1-1-onto-> B
35 f1ococnv1 5663 . . . . . . . . . 10  |-  ( ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) : ( B  ^m  1o ) -1-1-onto-> B  ->  ( `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  =  (  _I  |`  ( B  ^m  1o ) ) )
3634, 35mp1i 12 . . . . . . . . 9  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  =  (  _I  |`  ( B  ^m  1o ) ) )
3733, 36syl5eqr 2450 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( z  e.  B  |->  ( 1o 
X.  { z } ) )  o.  (
x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  =  (  _I  |`  ( B  ^m  1o ) ) )
3837coeq2d 4994 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  y )  o.  (
( z  e.  B  |->  ( 1o  X.  {
z } ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )  =  ( ( ( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) ) )
3926, 38syl5eq 2448 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( ( 1o eval  R ) `
 y )  o.  ( z  e.  B  |->  ( 1o  X.  {
z } ) ) )  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( ( ( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) ) )
40 eqid 2404 . . . . . . . 8  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
41 eqid 2404 . . . . . . . 8  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
42 simpl 444 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  R  e.  CRing )
43 ovex 6065 . . . . . . . . 9  |-  ( B  ^m  1o )  e. 
_V
4443a1i 11 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( B  ^m  1o )  e.  _V )
45 1on 6690 . . . . . . . . . . 11  |-  1o  e.  On
4620, 8, 21, 40evlrhm 19899 . . . . . . . . . . 11  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4745, 46mpan 652 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4823, 41rhmf 15782 . . . . . . . . . 10  |-  ( ( 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 ) ) ) )
4947, 48syl 16 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
5049ffvelrnda 5829 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
5140, 8, 41, 42, 44, 50pwselbas 13666 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
) : ( B  ^m  1o ) --> B )
52 fcoi1 5576 . . . . . . 7  |-  ( ( ( 1o eval  R ) `
 y ) : ( B  ^m  1o )
--> B  ->  ( (
( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) )  =  ( ( 1o eval  R
) `  y )
)
5351, 52syl 16 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) )  =  ( ( 1o eval  R
) `  y )
)
5425, 39, 533eqtrd 2440 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( ( 1o eval  R ) `  y
) )
55 ffn 5550 . . . . . . . 8  |-  ( ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
5649, 55syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
57 fnfvelrn 5826 . . . . . . 7  |-  ( ( ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ran  ( 1o eval  R ) )
5856, 57sylan 458 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ran  ( 1o eval  R ) )
59 mpfpf1.q . . . . . 6  |-  E  =  ran  ( 1o eval  R
)
6058, 59syl6eleqr 2495 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  E )
6154, 60eqeltrd 2478 . . . 4  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  e.  E )
62 coeq1 4989 . . . . 5  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
6362eleq1d 2470 . . . 4  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( ( (eval1 `  R ) `  y )  o.  (
x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E  <->  ( F  o.  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) ) )  e.  E ) )
6461, 63syl5ibcom 212 . . 3  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  =  F  -> 
( F  o.  (
x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E ) )
6564rexlimdva 2790 . 2  |-  ( R  e.  CRing  ->  ( E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E ) )
662, 19, 65sylc 58 1  |-  ( F  e.  Q  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916   (/)c0 3588   {csn 3774    e. cmpt 4226    _I cid 4453   Oncon0 4541    X. cxp 4835   `'ccnv 4836   ran crn 4838    |` cres 4839    o. ccom 4841    Fn wfn 5408   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   1oc1o 6676    ^m cmap 6977   Basecbs 13424    ^s cpws 13625   CRingccrg 15616   RingHom crh 15772   mPoly cmpl 16363   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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