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Theorem pf1mpf 18235
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 18232 . 2  |-  ( F  e.  Q  ->  R  e.  CRing )
3 id 22 . . . 4  |-  ( F  e.  Q  ->  F  e.  Q )
43, 1syl6eleq 2565 . . 3  |-  ( F  e.  Q  ->  F  e.  ran  (eval1 `  R ) )
5 eqid 2467 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
6 eqid 2467 . . . . . 6  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
7 eqid 2467 . . . . . 6  |-  ( R  ^s  B )  =  ( R  ^s  B )
8 pf1f.b . . . . . 6  |-  B  =  ( Base `  R
)
95, 6, 7, 8evl1rhm 18215 . . . . 5  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
102, 9syl 16 . . . 4  |-  ( F  e.  Q  ->  (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
11 eqid 2467 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
12 eqid 2467 . . . . 5  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
1311, 12rhmf 17224 . . . 4  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
14 ffn 5736 . . . 4  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
15 fvelrnb 5920 . . . 4  |-  ( (eval1 `  R )  Fn  ( Base `  (Poly1 `  R ) )  ->  ( F  e. 
ran  (eval1 `  R )  <->  E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F ) )
1610, 13, 14, 154syl 21 . . 3  |-  ( F  e.  Q  ->  ( F  e.  ran  (eval1 `  R
)  <->  E. y  e.  (
Base `  (Poly1 `  R
) ) ( (eval1 `  R ) `  y
)  =  F ) )
174, 16mpbid 210 . 2  |-  ( F  e.  Q  ->  E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F )
18 eqid 2467 . . . . . . . 8  |-  ( 1o eval  R )  =  ( 1o eval  R )
19 eqid 2467 . . . . . . . 8  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
20 eqid 2467 . . . . . . . . 9  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
216, 20, 11ply1bas 18081 . . . . . . . 8  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
225, 18, 8, 19, 21evl1val 18212 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  y )  =  ( ( ( 1o eval  R ) `  y )  o.  (
z  e.  B  |->  ( 1o  X.  { z } ) ) ) )
2322coeq1d 5169 . . . . . 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 `
 (/) ) ) ) )
24 coass 5531 . . . . . . 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 `  (/) ) ) ) )
25 df1o2 7152 . . . . . . . . . . 11  |-  1o  =  { (/) }
26 fvex 5881 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
278, 26eqeltri 2551 . . . . . . . . . . 11  |-  B  e. 
_V
28 0ex 4582 . . . . . . . . . . 11  |-  (/)  e.  _V
29 eqid 2467 . . . . . . . . . . 11  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) )  =  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )
3025, 27, 28, 29mapsncnv 7475 . . . . . . . . . 10  |-  `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  =  ( z  e.  B  |->  ( 1o  X.  { z } ) )
3130coeq1i 5167 . . . . . . . . 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 `
 (/) ) ) )
3225, 27, 28, 29mapsnf1o2 7476 . . . . . . . . . 10  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) : ( B  ^m  1o )
-1-1-onto-> B
33 f1ococnv1 5849 . . . . . . . . . 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 ) ) )
3432, 33mp1i 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 ) ) )
3531, 34syl5eqr 2522 . . . . . . . 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 ) ) )
3635coeq2d 5170 . . . . . . 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 ) ) ) )
3724, 36syl5eq 2520 . . . . . 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 ) ) ) )
38 eqid 2467 . . . . . . . 8  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
39 eqid 2467 . . . . . . . 8  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
40 simpl 457 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  R  e.  CRing )
41 ovex 6319 . . . . . . . . 9  |-  ( B  ^m  1o )  e. 
_V
4241a1i 11 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( B  ^m  1o )  e.  _V )
43 1on 7147 . . . . . . . . . . 11  |-  1o  e.  On
4418, 8, 19, 38evlrhm 18041 . . . . . . . . . . 11  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4543, 44mpan 670 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4621, 39rhmf 17224 . . . . . . . . . 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 ) ) ) )
4745, 46syl 16 . . . . . . . . 9  |-  ( R  e.  CRing  ->  ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
4847ffvelrnda 6031 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ( Base `  ( R  ^s  ( B  ^m  1o ) ) ) )
4938, 8, 39, 40, 42, 48pwselbas 14756 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
) : ( B  ^m  1o ) --> B )
50 fcoi1 5764 . . . . . . 7  |-  ( ( ( 1o eval  R ) `
 y ) : ( B  ^m  1o )
--> B  ->  ( (
( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) )  =  ( ( 1o eval  R
) `  y )
)
5149, 50syl 16 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( ( 1o eval  R ) `  y )  o.  (  _I  |`  ( B  ^m  1o ) ) )  =  ( ( 1o eval  R
) `  y )
)
5223, 37, 513eqtrd 2512 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( ( 1o eval  R ) `  y
) )
53 ffn 5736 . . . . . . . 8  |-  ( ( 1o eval  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  ( B  ^m  1o ) ) )  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
5447, 53syl 16 . . . . . . 7  |-  ( R  e.  CRing  ->  ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) ) )
55 fnfvelrn 6028 . . . . . . 7  |-  ( ( ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ran  ( 1o eval  R ) )
5654, 55sylan 471 . . . . . 6  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ran  ( 1o eval  R ) )
57 mpfpf1.q . . . . . 6  |-  E  =  ran  ( 1o eval  R
)
5856, 57syl6eleqr 2566 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  E )
5952, 58eqeltrd 2555 . . . 4  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  e.  E )
60 coeq1 5165 . . . . 5  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
6160eleq1d 2536 . . . 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 ) )
6259, 61syl5ibcom 220 . . 3  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  =  F  -> 
( F  o.  (
x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E ) )
6362rexlimdva 2959 . 2  |-  ( R  e.  CRing  ->  ( E. y  e.  ( Base `  (Poly1 `  R ) ) ( (eval1 `  R ) `  y )  =  F  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E ) )
642, 17, 63sylc 60 1  |-  ( F  e.  Q  ->  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) )  e.  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   (/)c0 3790   {csn 4032    |-> cmpt 4510    _I cid 4795   Oncon0 4883    X. cxp 5002   `'ccnv 5003   ran crn 5005    |` cres 5006    o. ccom 5008    Fn wfn 5588   -->wf 5589   -1-1-onto->wf1o 5592   ` cfv 5593  (class class class)co 6294   1oc1o 7133    ^m cmap 7430   Basecbs 14502    ^s cpws 14714   CRingccrg 17048   RingHom crh 17210   mPoly cmpl 17849   eval cevl 18017  PwSer1cps1 18061  Poly1cpl1 18063  eval1ce1 18198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-ofr 6535  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-1o 7140  df-2o 7141  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-hash 12384  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-ip 14585  df-tset 14586  df-ple 14587  df-ds 14589  df-hom 14591  df-cco 14592  df-0g 14709  df-gsum 14710  df-prds 14715  df-pws 14717  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-mulg 15909  df-subg 16047  df-ghm 16114  df-cntz 16204  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-srg 17007  df-ring 17049  df-cring 17050  df-rnghom 17213  df-subrg 17275  df-lmod 17362  df-lss 17427  df-lsp 17466  df-assa 17808  df-asp 17809  df-ascl 17810  df-psr 17852  df-mvr 17853  df-mpl 17854  df-opsr 17856  df-evls 18018  df-evl 18019  df-psr1 18066  df-ply1 18068  df-evl1 18200
This theorem is referenced by:  pf1ind  18238
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