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Theorem mzpmfp 26825
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 10032 . . . . . . 7  |-  ZZ  C_  CC
2 eqid 2283 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 cnfldbas 16383 . . . . . . . 8  |-  CC  =  ( Base ` fld )
42, 3ressbas2 13199 . . . . . . 7  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
51, 4ax-mp 8 . . . . . 6  |-  ZZ  =  ( Base `  (flds  ZZ ) )
6 eqid 2283 . . . . . . . 8  |-  ( I eval  (flds  ZZ ) )  =  ( I eval  (flds  ZZ ) )
76, 5evlval 19408 . . . . . . 7  |-  ( I eval  (flds  ZZ ) )  =  ( ( I evalSub  (flds  ZZ ) ) `  ZZ )
87rneqi 4905 . . . . . 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 16395 . . . . . . . 8  |-fld  e.  CRing
11 zsubrg 16425 . . . . . . . 8  |-  ZZ  e.  (SubRing ` fld )
122subrgcrng 15549 . . . . . . . 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 15548 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1611, 15ax-mp 8 . . . . . . . 8  |-  (flds  ZZ )  e.  Ring
175subrgid 15547 . . . . . . . 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 19422 . . . . 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 19423 . . . . 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 10033 . . . . . . . 8  |-  ZZ  e.  _V
30 cnfldadd 16384 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
312, 30ressplusg 13250 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  +  =  ( +g  `  (flds  ZZ )
) )
3229, 31ax-mp 8 . . . . . . 7  |-  +  =  ( +g  `  (flds  ZZ ) )
338, 32mpfaddcl 19426 . . . . . 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 16385 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
362, 35ressmulr 13261 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3729, 36ax-mp 8 . . . . . . 7  |-  x.  =  ( .r `  (flds  ZZ ) )
388, 37mpfmulcl 19427 . . . . . 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 2343 . . . . 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 2343 . . . . 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 2343 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  f  e.  ran  ( I eval  (flds  ZZ ) ) ) )
43 eleq1 2343 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )
44 eleq1 2343 . . . . 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 2343 . . . . 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 2343 . . . . 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 26824 . . . 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 26816 . . . . . 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 26817 . . . . . 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 2343 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
55 eleq1 2343 . . . . 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 2343 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
57 eleq1 2343 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
58 eleq1 2343 . . . . 5  |-  ( b  =  ( x  o F  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  +  y )  e.  (mzPoly `  I ) ) )
59 eleq1 2343 . . . . 5  |-  ( b  =  ( x  o F  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  x.  y )  e.  (mzPoly `  I ) ) )
60 eleq1 2343 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
61 mzpconst 26813 . . . . . 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 26815 . . . . . 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 19428 . . . 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 2281 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
69 fvprc 5519 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
70 df-evl 16102 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
7170reldmmpt2 5955 . . . . . 6  |-  Rel  dom eval
7271ovprc1 5886 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval  (flds  ZZ ) )  =  (/) )
7372rneqd 4906 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  ran  (/) )
74 rn0 4936 . . . 4  |-  ran  (/)  =  (/)
7573, 74syl6eq 2331 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  (/) )
7669, 75eqtr4d 2318 . 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 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076    ^m cmap 6772   CCcc 8735    + caddc 8740    x. cmul 8742   ZZcz 10024   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   Ringcrg 15337   CRingccrg 15338  SubRingcsubrg 15541   evalSub ces 16090   eval cevl 16091  ℂfldccnfld 16377  mzPolycmzp 26800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-rnghom 15496  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lsp 15729  df-assa 16053  df-asp 16054  df-ascl 16055  df-psr 16098  df-mvr 16099  df-mpl 16100  df-evls 16101  df-evl 16102  df-cnfld 16378  df-mzpcl 26801  df-mzp 26802
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