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Theorem pf1mpf 17789
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 17786 . 2  |-  ( F  e.  Q  ->  R  e.  CRing )
3 id 22 . . . 4  |-  ( F  e.  Q  ->  F  e.  Q )
43, 1syl6eleq 2533 . . 3  |-  ( F  e.  Q  ->  F  e.  ran  (eval1 `  R ) )
5 eqid 2443 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
6 eqid 2443 . . . . . 6  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
7 eqid 2443 . . . . . 6  |-  ( R  ^s  B )  =  ( R  ^s  B )
8 pf1f.b . . . . . 6  |-  B  =  ( Base `  R
)
95, 6, 7, 8evl1rhm 17769 . . . . 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 2443 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
12 eqid 2443 . . . . 5  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
1311, 12rhmf 16819 . . . 4  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
14 ffn 5562 . . . 4  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
15 fvelrnb 5742 . . . 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 2443 . . . . . . . 8  |-  ( 1o eval  R )  =  ( 1o eval  R )
19 eqid 2443 . . . . . . . 8  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
20 eqid 2443 . . . . . . . . 9  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
216, 20, 11ply1bas 17654 . . . . . . . 8  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
225, 18, 8, 19, 21evl1val 17766 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  y )  =  ( ( ( 1o eval  R ) `  y )  o.  (
z  e.  B  |->  ( 1o  X.  { z } ) ) ) )
2322coeq1d 5004 . . . . . 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 5359 . . . . . . 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 6935 . . . . . . . . . . 11  |-  1o  =  { (/) }
26 fvex 5704 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
278, 26eqeltri 2513 . . . . . . . . . . 11  |-  B  e. 
_V
28 0ex 4425 . . . . . . . . . . 11  |-  (/)  e.  _V
29 eqid 2443 . . . . . . . . . . 11  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) )  =  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )
3025, 27, 28, 29mapsncnv 7262 . . . . . . . . . 10  |-  `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  =  ( z  e.  B  |->  ( 1o  X.  { z } ) )
3130coeq1i 5002 . . . . . . . . 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 7263 . . . . . . . . . 10  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) : ( B  ^m  1o )
-1-1-onto-> B
33 f1ococnv1 5672 . . . . . . . . . 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 2489 . . . . . . . 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 5005 . . . . . . 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 2487 . . . . . 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 2443 . . . . . . . 8  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
39 eqid 2443 . . . . . . . 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 6119 . . . . . . . . 9  |-  ( B  ^m  1o )  e. 
_V
4241a1i 11 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( B  ^m  1o )  e.  _V )
43 1on 6930 . . . . . . . . . . 11  |-  1o  e.  On
4418, 8, 19, 38evlrhm 17614 . . . . . . . . . . 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 16819 . . . . . . . . . 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 5846 . . . . . . . 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 14430 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
) : ( B  ^m  1o ) --> B )
50 fcoi1 5588 . . . . . . 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 2479 . . . . 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 5562 . . . . . . . 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 5843 . . . . . . 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 2534 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  E )
5952, 58eqeltrd 2517 . . . 4  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  e.  E )
60 coeq1 5000 . . . . 5  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
6160eleq1d 2509 . . . 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 2844 . 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 1369    e. wcel 1756   E.wrex 2719   _Vcvv 2975   (/)c0 3640   {csn 3880    e. cmpt 4353    _I cid 4634   Oncon0 4722    X. cxp 4841   `'ccnv 4842   ran crn 4844    |` cres 4845    o. ccom 4847    Fn wfn 5416   -->wf 5417   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6094   1oc1o 6916    ^m cmap 7217   Basecbs 14177    ^s cpws 14388   CRingccrg 16649   RingHom crh 16807   mPoly cmpl 17423   eval cevl 17590  PwSer1cps1 17634  Poly1cpl1 17636  eval1ce1 17752
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 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-inf2 7850  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
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 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-of 6323  df-ofr 6324  df-om 6480  df-1st 6580  df-2nd 6581  df-supp 6694  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-pm 7220  df-ixp 7267  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-fsupp 7624  df-sup 7694  df-oi 7727  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-fz 11441  df-fzo 11552  df-seq 11810  df-hash 12107  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-sca 14257  df-vsca 14258  df-ip 14259  df-tset 14260  df-ple 14261  df-ds 14263  df-hom 14265  df-cco 14266  df-0g 14383  df-gsum 14384  df-prds 14389  df-pws 14391  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-mhm 15467  df-submnd 15468  df-grp 15548  df-minusg 15549  df-sbg 15550  df-mulg 15551  df-subg 15681  df-ghm 15748  df-cntz 15838  df-cmn 16282  df-abl 16283  df-mgp 16595  df-ur 16607  df-srg 16611  df-rng 16650  df-cring 16651  df-rnghom 16809  df-subrg 16866  df-lmod 16953  df-lss 17017  df-lsp 17056  df-assa 17387  df-asp 17388  df-ascl 17389  df-psr 17426  df-mvr 17427  df-mpl 17428  df-opsr 17430  df-evls 17591  df-evl 17592  df-psr1 17639  df-ply1 17641  df-evl1 17754
This theorem is referenced by:  pf1ind  17792
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