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Theorem mzpmfp 26804
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 10290 . . . . . . 7  |-  ZZ  C_  CC
2 eqid 2436 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 cnfldbas 16707 . . . . . . . 8  |-  CC  =  ( Base ` fld )
42, 3ressbas2 13520 . . . . . . 7  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
51, 4ax-mp 8 . . . . . 6  |-  ZZ  =  ( Base `  (flds  ZZ ) )
6 eqid 2436 . . . . . . . 8  |-  ( I eval  (flds  ZZ ) )  =  ( I eval  (flds  ZZ ) )
76, 5evlval 19945 . . . . . . 7  |-  ( I eval  (flds  ZZ ) )  =  ( ( I evalSub  (flds  ZZ ) ) `  ZZ )
87rneqi 5096 . . . . . 6  |-  ran  (
I eval  (flds  ZZ ) )  =  ran  ( ( I evalSub  (flds  ZZ )
) `  ZZ )
9 simpl 444 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  I  e.  _V )
10 cncrng 16722 . . . . . . . 8  |-fld  e.  CRing
11 zsubrg 16752 . . . . . . . 8  |-  ZZ  e.  (SubRing ` fld )
122subrgcrng 15872 . . . . . . . 8  |-  ( (fld  e. 
CRing  /\  ZZ  e.  (SubRing ` fld ) )  ->  (flds  ZZ )  e.  CRing )
1310, 11, 12mp2an 654 . . . . . . 7  |-  (flds  ZZ )  e.  CRing
1413a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  (flds  ZZ )  e.  CRing )
152subrgrng 15871 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1611, 15ax-mp 8 . . . . . . . 8  |-  (flds  ZZ )  e.  Ring
175subrgid 15870 . . . . . . . 8  |-  ( (flds  ZZ )  e.  Ring  ->  ZZ  e.  (SubRing `  (flds  ZZ ) ) )
1816, 17ax-mp 8 . . . . . . 7  |-  ZZ  e.  (SubRing `  (flds  ZZ ) )
1918a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ZZ  e.  (SubRing `  (flds  ZZ )
) )
20 simpr 448 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  f  e.  ZZ )
215, 8, 9, 14, 19, 20mpfconst 19959 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
f } )  e. 
ran  ( I eval  (flds  ZZ )
) )
22 simpl 444 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  I  e.  _V )
2313a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  (flds  ZZ )  e.  CRing )
2418a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ZZ  e.  (SubRing `  (flds  ZZ )
) )
25 simpr 448 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  f  e.  I )
265, 8, 22, 23, 24, 25mpfproj 19960 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval  (flds  ZZ ) ) )
27 simp2r 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 ) ) ) )  ->  f  e.  ran  ( I eval  (flds  ZZ ) ) )
28 simp3r 986 . . . . . 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 10291 . . . . . . . 8  |-  ZZ  e.  _V
30 cnfldadd 16708 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
312, 30ressplusg 13571 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  +  =  ( +g  `  (flds  ZZ )
) )
3229, 31ax-mp 8 . . . . . . 7  |-  +  =  ( +g  `  (flds  ZZ ) )
338, 32mpfaddcl 19963 . . . . . 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 643 . . . . 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 16709 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
362, 35ressmulr 13582 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3729, 36ax-mp 8 . . . . . . 7  |-  x.  =  ( .r `  (flds  ZZ ) )
388, 37mpfmulcl 19964 . . . . . 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 643 . . . . 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 2496 . . . . 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 2496 . . . . 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 2496 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  f  e.  ran  ( I eval  (flds  ZZ ) ) ) )
43 eleq1 2496 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )
44 eleq1 2496 . . . . 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 2496 . . . . 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 2496 . . . . 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 26803 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval  (flds  ZZ )
) )
48 simprlr 740 . . . . . 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 742 . . . . . 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 26795 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  o F  +  y
)  e.  (mzPoly `  I ) )
5148, 49, 50syl2anc 643 . . . . 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 26796 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  o F  x.  y
)  e.  (mzPoly `  I ) )
5348, 49, 52syl2anc 643 . . . . 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 2496 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
55 eleq1 2496 . . . . 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 2496 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
57 eleq1 2496 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
58 eleq1 2496 . . . . 5  |-  ( b  =  ( x  o F  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  +  y )  e.  (mzPoly `  I ) ) )
59 eleq1 2496 . . . . 5  |-  ( b  =  ( x  o F  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  o F  x.  y )  e.  (mzPoly `  I ) ) )
60 eleq1 2496 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
61 mzpconst 26792 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
6261adantlr 696 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
63 mzpproj 26794 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
6463adantlr 696 . . . . 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 448 . . . . 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 19965 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval  (flds  ZZ ) ) )  -> 
a  e.  (mzPoly `  I ) )
6747, 66impbida 806 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval  (flds  ZZ ) ) ) )
6867eqrdv 2434 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
69 fvprc 5722 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
70 df-evl 16421 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
7170reldmmpt2 6181 . . . . . 6  |-  Rel  dom eval
7271ovprc1 6109 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval  (flds  ZZ ) )  =  (/) )
7372rneqd 5097 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  ran  (/) )
74 rn0 5127 . . . 4  |-  ran  (/)  =  (/)
7573, 74syl6eq 2484 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  (/) )
7669, 75eqtr4d 2471 . 2  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
7768, 76pm2.61i 158 1  |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   (/)c0 3628   {csn 3814    e. cmpt 4266    X. cxp 4876   ran crn 4879   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303    ^m cmap 7018   CCcc 8988    + caddc 8993    x. cmul 8995   ZZcz 10282   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   .rcmulr 13530   Ringcrg 15660   CRingccrg 15661  SubRingcsubrg 15864   evalSub ces 16409   eval cevl 16410  ℂfldccnfld 16703  mzPolycmzp 26779
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-rnghom 15819  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-assa 16372  df-asp 16373  df-ascl 16374  df-psr 16417  df-mvr 16418  df-mpl 16419  df-evls 16420  df-evl 16421  df-cnfld 16704  df-mzpcl 26780  df-mzp 26781
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