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Theorem pf1mpf 21403
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 21400 . 2  |-  ( F  e.  Q  ->  R  e.  CRing )
3 id 22 . . . 4  |-  ( F  e.  Q  ->  F  e.  Q )
43, 1syl6eleq 2523 . . 3  |-  ( F  e.  Q  ->  F  e.  ran  (eval1 `  R ) )
5 eqid 2433 . . . . . 6  |-  (eval1 `  R
)  =  (eval1 `  R
)
6 eqid 2433 . . . . . 6  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
7 eqid 2433 . . . . . 6  |-  ( R  ^s  B )  =  ( R  ^s  B )
8 pf1f.b . . . . . 6  |-  B  =  ( Base `  R
)
95, 6, 7, 8evl1rhm 21380 . . . . 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 2433 . . . . 5  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
12 eqid 2433 . . . . 5  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
1311, 12rhmf 16748 . . . 4  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
14 ffn 5547 . . . 4  |-  ( (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) )  ->  (eval1 `  R
)  Fn  ( Base `  (Poly1 `  R ) ) )
15 fvelrnb 5727 . . . 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 2433 . . . . . . . 8  |-  ( 1o eval  R )  =  ( 1o eval  R )
19 eqid 2433 . . . . . . . 8  |-  ( 1o mPoly  R )  =  ( 1o mPoly  R )
20 eqid 2433 . . . . . . . . 9  |-  (PwSer1 `  R
)  =  (PwSer1 `  R
)
216, 20, 11ply1bas 17550 . . . . . . . 8  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  ( 1o mPoly  R ) )
225, 18, 8, 19, 21evl1val 21379 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  y )  =  ( ( ( 1o eval  R ) `  y )  o.  (
z  e.  B  |->  ( 1o  X.  { z } ) ) ) )
2322coeq1d 4988 . . . . . 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 5344 . . . . . . 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 6920 . . . . . . . . . . 11  |-  1o  =  { (/) }
26 fvex 5689 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
278, 26eqeltri 2503 . . . . . . . . . . 11  |-  B  e. 
_V
28 0ex 4410 . . . . . . . . . . 11  |-  (/)  e.  _V
29 eqid 2433 . . . . . . . . . . 11  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) )  =  ( x  e.  ( B  ^m  1o ) 
|->  ( x `  (/) ) )
3025, 27, 28, 29mapsncnv 7247 . . . . . . . . . 10  |-  `' ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) )  =  ( z  e.  B  |->  ( 1o  X.  { z } ) )
3130coeq1i 4986 . . . . . . . . 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 7248 . . . . . . . . . 10  |-  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) : ( B  ^m  1o )
-1-1-onto-> B
33 f1ococnv1 5657 . . . . . . . . . 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 2479 . . . . . . . 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 4989 . . . . . . 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 2477 . . . . . 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 2433 . . . . . . . 8  |-  ( R  ^s  ( B  ^m  1o ) )  =  ( R  ^s  ( B  ^m  1o ) )
39 eqid 2433 . . . . . . . 8  |-  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )  =  ( Base `  ( R  ^s  ( B  ^m  1o ) ) )
40 simpl 454 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  R  e.  CRing )
41 ovex 6105 . . . . . . . . 9  |-  ( B  ^m  1o )  e. 
_V
4241a1i 11 . . . . . . . 8  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( B  ^m  1o )  e.  _V )
43 1on 6915 . . . . . . . . . . 11  |-  1o  e.  On
4418, 8, 19, 38evlrhm 21377 . . . . . . . . . . 11  |-  ( ( 1o  e.  On  /\  R  e.  CRing )  -> 
( 1o eval  R )  e.  ( ( 1o mPoly  R
) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4543, 44mpan 663 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  ( 1o eval  R )  e.  ( ( 1o mPoly  R ) RingHom  ( R  ^s  ( B  ^m  1o ) ) ) )
4621, 39rhmf 16748 . . . . . . . . . 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 5831 . . . . . . . 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 14410 . . . . . . 7  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
) : ( B  ^m  1o ) --> B )
50 fcoi1 5573 . . . . . . 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 2469 . . . . 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 5547 . . . . . . . 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 5828 . . . . . . 7  |-  ( ( ( 1o eval  R )  Fn  ( Base `  (Poly1 `  R ) )  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  ran  ( 1o eval  R ) )
5654, 55sylan 468 . . . . . 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 2524 . . . . 5  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( 1o eval  R ) `  y
)  e.  E )
5952, 58eqeltrd 2507 . . . 4  |-  ( ( R  e.  CRing  /\  y  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  e.  E )
60 coeq1 4984 . . . . 5  |-  ( ( (eval1 `  R ) `  y )  =  F  ->  ( ( (eval1 `  R ) `  y
)  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `
 (/) ) ) )  =  ( F  o.  ( x  e.  ( B  ^m  1o )  |->  ( x `  (/) ) ) ) )
6160eleq1d 2499 . . . 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 2831 . 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 1362    e. wcel 1755   E.wrex 2706   _Vcvv 2962   (/)c0 3625   {csn 3865    e. cmpt 4338    _I cid 4618   Oncon0 4706    X. cxp 4825   `'ccnv 4826   ran crn 4828    |` cres 4829    o. ccom 4831    Fn wfn 5401   -->wf 5402   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   1oc1o 6901    ^m cmap 7202   Basecbs 14157    ^s cpws 14368   CRingccrg 16578   RingHom crh 16738   mPoly cmpl 17342   eval cevl 17344  PwSer1cps1 17526  Poly1cpl1 17528  eval1ce1 17530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-ofr 6310  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-hom 14245  df-cco 14246  df-0g 14363  df-gsum 14364  df-prds 14369  df-pws 14371  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-mhm 15447  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-ghm 15725  df-cntz 15815  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-rnghom 16740  df-subrg 16787  df-lmod 16874  df-lss 16936  df-lsp 16975  df-assa 17306  df-asp 17307  df-ascl 17308  df-psr 17351  df-mvr 17352  df-mpl 17353  df-evls 17354  df-evl 17355  df-opsr 17359  df-psr1 17533  df-ply1 17535  df-evl1 17537
This theorem is referenced by:  pf1ind  21406
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