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Theorem pf1mpf 19960
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 19957 . 2  |-  ( F  e.  Q  ->  R  e.  CRing )
3 id 20 . . . 4  |-  ( F  e.  Q  ->  F  e.  Q )
43, 1syl6eleq 2525 . . 3  |-  ( F  e.  Q  ->  F  e.  ran  (eval1 `  R ) )
5 eqid 2435 . . . . . . . 8  |-  (eval1 `  R
)  =  (eval1 `  R
)
6 eqid 2435 . . . . . . . 8  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
7 eqid 2435 . . . . . . . 8  |-  ( R  ^s  B )  =  ( R  ^s  B )
8 pf1f.b . . . . . . . 8  |-  B  =  ( Base `  R
)
95, 6, 7, 8evl1rhm 19937 . . . . . . 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 2435 . . . . . . 7  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
12 eqid 2435 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
1311, 12rhmf 15815 . . . . . 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 5582 . . . . 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 5765 . . . 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 2435 . . . . . . . 8  |-  ( 1o eval  R )  =  ( 1o eval  R )
21 eqid 2435 . . . . . . . 8  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
22 eqid 2435 . . . . . . . . 9  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
236, 22, 11ply1bas 16581 . . . . . . . 8  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
245, 20, 8, 21, 23evl1val 19936 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  y )  =  ( ( ( 1o eval  R ) `  y )  o.  (
z  e.  B  |->  ( 1o  X.  { z } ) ) ) )
2524coeq1d 5025 . . . . . 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 5379 . . . . . . 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 6727 . . . . . . . . . . 11  |-  1o  =  { (/) }
28 fvex 5733 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
298, 28eqeltri 2505 . . . . . . . . . . 11  |-  B  e. 
_V
30 0ex 4331 . . . . . . . . . . 11  |-  (/)  e.  _V
31 eqid 2435 . . . . . . . . . . 11  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) )  =  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )
3227, 29, 30, 31mapsncnv 7051 . . . . . . . . . 10  |-  `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  =  ( z  e.  B  |->  ( 1o  X.  { z } ) )
3332coeq1i 5023 . . . . . . . . 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 7052 . . . . . . . . . 10  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) : ( B  ^m  1o )
-1-1-onto-> B
35 f1ococnv1 5695 . . . . . . . . . 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 2481 . . . . . . . 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 5026 . . . . . . 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 2479 . . . . . 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 2435 . . . . . . . 8  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
41 eqid 2435 . . . . . . . 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 6097 . . . . . . . . 9  |-  ( B  ^m  1o )  e. 
_V
4443a1i 11 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( B  ^m  1o )  e.  _V )
45 1on 6722 . . . . . . . . . . 11  |-  1o  e.  On
4620, 8, 21, 40evlrhm 19934 . . . . . . . . . . 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 15815 . . . . . . . . . 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 5861 . . . . . . . 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 13699 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
) : ( B  ^m  1o ) --> B )
52 fcoi1 5608 . . . . . . 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 2471 . . . . 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 5582 . . . . . . . 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 5858 . . . . . . 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 2526 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  E )
6154, 60eqeltrd 2509 . . . 4  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  e.  E )
62 coeq1 5021 . . . . 5  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
6362eleq1d 2501 . . . 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 2822 . 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 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948   (/)c0 3620   {csn 3806    e. cmpt 4258    _I cid 4485   Oncon0 4573    X. cxp 4867   `'ccnv 4868   ran crn 4870    |` cres 4871    o. ccom 4873    Fn wfn 5440   -->wf 5441   -1-1-onto->wf1o 5444   ` cfv 5445  (class class class)co 6072   1oc1o 6708    ^m cmap 7009   Basecbs 13457    ^s cpws 13658   CRingccrg 15649   RingHom crh 15805   mPoly cmpl 16396   eval cevl 16398  PwSer1cps1 16557  Poly1cpl1 16559  eval1ce1 16561
This theorem is referenced by:  pf1ind  19963
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-ofr 6297  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-sup 7437  df-oi 7468  df-card 7815  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-fz 11033  df-fzo 11124  df-seq 11312  df-hash 11607  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-hom 13541  df-cco 13542  df-prds 13659  df-pws 13661  df-0g 13715  df-gsum 13716  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-mhm 14726  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-mulg 14803  df-subg 14929  df-ghm 14992  df-cntz 15104  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-cring 15652  df-ur 15653  df-rnghom 15807  df-subrg 15854  df-lmod 15940  df-lss 15997  df-lsp 16036  df-assa 16360  df-asp 16361  df-ascl 16362  df-psr 16405  df-mvr 16406  df-mpl 16407  df-evls 16408  df-evl 16409  df-opsr 16413  df-psr1 16564  df-ply1 16566  df-evl1 16568
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