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Theorem mzpmfp 26928
Description: Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.)
Assertion
Ref Expression
mzpmfp  |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )

Proof of Theorem mzpmfp
Dummy variables  a 
b  x  y  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsscn 10048 . . . . . . 7  |-  ZZ  C_  CC
2 eqid 2296 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 cnfldbas 16399 . . . . . . . 8  |-  CC  =  ( Base ` fld )
42, 3ressbas2 13215 . . . . . . 7  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
51, 4ax-mp 8 . . . . . 6  |-  ZZ  =  ( Base `  (flds  ZZ ) )
6 eqid 2296 . . . . . . . 8  |-  ( I eval  (flds  ZZ ) )  =  ( I eval  (flds  ZZ ) )
76, 5evlval 19424 . . . . . . 7  |-  ( I eval  (flds  ZZ ) )  =  ( ( I evalSub  (flds  ZZ ) ) `  ZZ )
87rneqi 4921 . . . . . 6  |-  ran  (
I eval  (flds  ZZ ) )  =  ran  ( ( I evalSub  (flds  ZZ )
) `  ZZ )
9 simpl 443 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  I  e.  _V )
10 cncrng 16411 . . . . . . . 8  |-fld  e.  CRing
11 zsubrg 16441 . . . . . . . 8  |-  ZZ  e.  (SubRing ` fld )
122subrgcrng 15565 . . . . . . . 8  |-  ( (fld  e. 
CRing  /\  ZZ  e.  (SubRing ` fld ) )  ->  (flds  ZZ )  e.  CRing )
1310, 11, 12mp2an 653 . . . . . . 7  |-  (flds  ZZ )  e.  CRing
1413a1i 10 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  (flds  ZZ )  e.  CRing )
152subrgrng 15564 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1611, 15ax-mp 8 . . . . . . . 8  |-  (flds  ZZ )  e.  Ring
175subrgid 15563 . . . . . . . 8  |-  ( (flds  ZZ )  e.  Ring  ->  ZZ  e.  (SubRing `  (flds  ZZ ) ) )
1816, 17ax-mp 8 . . . . . . 7  |-  ZZ  e.  (SubRing `  (flds  ZZ ) )
1918a1i 10 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ZZ  e.  (SubRing `  (flds  ZZ )
) )
20 simpr 447 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  f  e.  ZZ )
215, 8, 9, 14, 19, 20mpfconst 19438 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
f } )  e. 
ran  ( I eval  (flds  ZZ )
) )
22 simpl 443 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  I  e.  _V )
2313a1i 10 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  (flds  ZZ )  e.  CRing )
2418a1i 10 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ZZ  e.  (SubRing `  (flds  ZZ )
) )
25 simpr 447 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  f  e.  I )
265, 8, 22, 23, 24, 25mpfproj 19439 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval  (flds  ZZ ) ) )
27 simp2r 982 . . . . . 6  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval  (flds  ZZ )
) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )  ->  f  e.  ran  ( I eval  (flds  ZZ ) ) )
28 simp3r 984 . . . . . 6  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval  (flds  ZZ )
) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )  ->  g  e.  ran  ( I eval  (flds  ZZ ) ) )
29 zex 10049 . . . . . . . 8  |-  ZZ  e.  _V
30 cnfldadd 16400 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
312, 30ressplusg 13266 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  +  =  ( +g  `  (flds  ZZ )
) )
3229, 31ax-mp 8 . . . . . . 7  |-  +  =  ( +g  `  (flds  ZZ ) )
338, 32mpfaddcl 19442 . . . . . 6  |-  ( ( f  e.  ran  (
I eval  (flds  ZZ ) )  /\  g  e.  ran  ( I eval  (flds  ZZ )
) )  ->  (
f  o F  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
3427, 28, 33syl2anc 642 . . . . 5  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval  (flds  ZZ )
) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )  ->  ( f  o F  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
35 cnfldmul 16401 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
362, 35ressmulr 13277 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3729, 36ax-mp 8 . . . . . . 7  |-  x.  =  ( .r `  (flds  ZZ ) )
388, 37mpfmulcl 19443 . . . . . 6  |-  ( ( f  e.  ran  (
I eval  (flds  ZZ ) )  /\  g  e.  ran  ( I eval  (flds  ZZ )
) )  ->  (
f  o F  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
3927, 28, 38syl2anc 642 . . . . 5  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval  (flds  ZZ )
) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )  ->  ( f  o F  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
40 eleq1 2356 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ f } )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( ( ZZ 
^m  I )  X. 
{ f } )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
41 eleq1 2356 . . . . 5  |-  ( b  =  ( g  e.  ( ZZ  ^m  I
)  |->  ( g `  f ) )  -> 
( b  e.  ran  ( I eval  (flds  ZZ ) )  <->  ( g  e.  ( ZZ  ^m  I
)  |->  ( g `  f ) )  e. 
ran  ( I eval  (flds  ZZ )
) ) )
42 eleq1 2356 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  f  e.  ran  ( I eval  (flds  ZZ ) ) ) )
43 eleq1 2356 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )
44 eleq1 2356 . . . . 5  |-  ( b  =  ( f  o F  +  g )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( f  o F  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
45 eleq1 2356 . . . . 5  |-  ( b  =  ( f  o F  x.  g )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( f  o F  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
46 eleq1 2356 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  a  e.  ran  ( I eval  (flds  ZZ ) ) ) )
4721, 26, 34, 39, 40, 41, 42, 43, 44, 45, 46mzpindd 26927 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval  (flds  ZZ )
) )
48 simprlr 739 . . . . . 6  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  ( ( x  e. 
ran  ( I eval  (flds  ZZ )
)  /\  x  e.  (mzPoly `  I ) )  /\  ( y  e. 
ran  ( I eval  (flds  ZZ )
)  /\  y  e.  (mzPoly `  I ) ) ) )  ->  x  e.  (mzPoly `  I )
)
49 simprrr 741 . . . . . 6  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  ( ( x  e. 
ran  ( I eval  (flds  ZZ )
)  /\  x  e.  (mzPoly `  I ) )  /\  ( y  e. 
ran  ( I eval  (flds  ZZ )
)  /\  y  e.  (mzPoly `  I ) ) ) )  ->  y  e.  (mzPoly `  I )
)
50 mzpadd 26919 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  o F  +  y
)  e.  (mzPoly `  I ) )
5148, 49, 50syl2anc 642 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  ( ( x  e. 
ran  ( I eval  (flds  ZZ )
)  /\  x  e.  (mzPoly `  I ) )  /\  ( y  e. 
ran  ( I eval  (flds  ZZ )
)  /\  y  e.  (mzPoly `  I ) ) ) )  ->  (
x  o F  +  y )  e.  (mzPoly `  I ) )
52 mzpmul 26920 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  o F  x.  y
)  e.  (mzPoly `  I ) )
5348, 49, 52syl2anc 642 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  ( ( x  e. 
ran  ( I eval  (flds  ZZ )
)  /\  x  e.  (mzPoly `  I ) )  /\  ( y  e. 
ran  ( I eval  (flds  ZZ )
)  /\  y  e.  (mzPoly `  I ) ) ) )  ->  (
x  o F  x.  y )  e.  (mzPoly `  I ) )
54 eleq1 2356 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
55 eleq1 2356 . . . . 5  |-  ( b  =  ( y  e.  ( ZZ  ^m  I
)  |->  ( y `  x ) )  -> 
( b  e.  (mzPoly `  I )  <->  ( y  e.  ( ZZ  ^m  I
)  |->  ( y `  x ) )  e.  (mzPoly `  I )
) )
56 eleq1 2356 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
57 eleq1 2356 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
58 eleq1 2356 . . . . 5  |-  ( b  =  ( x  o F  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  +  y )  e.  (mzPoly `  I ) ) )
59 eleq1 2356 . . . . 5  |-  ( b  =  ( x  o F  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  x.  y )  e.  (mzPoly `  I ) ) )
60 eleq1 2356 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
61 mzpconst 26916 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
6261adantlr 695 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
63 mzpproj 26918 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
6463adantlr 695 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
65 simpr 447 . . . . 5  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval  (flds  ZZ ) ) )  -> 
a  e.  ran  (
I eval  (flds  ZZ ) ) )
665, 32, 37, 8, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 64, 65mpfind 19444 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval  (flds  ZZ ) ) )  -> 
a  e.  (mzPoly `  I ) )
6747, 66impbida 805 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval  (flds  ZZ ) ) ) )
6867eqrdv 2294 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
69 fvprc 5535 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
70 df-evl 16118 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
7170reldmmpt2 5971 . . . . . 6  |-  Rel  dom eval
7271ovprc1 5902 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval  (flds  ZZ ) )  =  (/) )
7372rneqd 4922 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  ran  (/) )
74 rn0 4952 . . . 4  |-  ran  (/)  =  (/)
7573, 74syl6eq 2344 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  (/) )
7669, 75eqtr4d 2331 . 2  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
7768, 76pm2.61i 156 1  |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092    ^m cmap 6788   CCcc 8751    + caddc 8756    x. cmul 8758   ZZcz 10040   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   .rcmulr 13225   Ringcrg 15353   CRingccrg 15354  SubRingcsubrg 15557   evalSub ces 16106   eval cevl 16107  ℂfldccnfld 16393  mzPolycmzp 26903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117  df-evl 16118  df-cnfld 16394  df-mzpcl 26904  df-mzp 26905
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