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Theorem mzpmfpOLD 30614
Description: Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) Obsolete version of mzpmfp 30613 as of 13-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
mzpmfpOLD  |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )

Proof of Theorem mzpmfpOLD
Dummy variables  a 
b  x  y  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsscn 10884 . . . . . . 7  |-  ZZ  C_  CC
2 eqid 2467 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 cnfldbas 18292 . . . . . . . 8  |-  CC  =  ( Base ` fld )
42, 3ressbas2 14562 . . . . . . 7  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
51, 4ax-mp 5 . . . . . 6  |-  ZZ  =  ( Base `  (flds  ZZ ) )
6 eqid 2467 . . . . . . . 8  |-  ( I eval  (flds  ZZ ) )  =  ( I eval  (flds  ZZ ) )
76, 5evlval 18061 . . . . . . 7  |-  ( I eval  (flds  ZZ ) )  =  ( ( I evalSub  (flds  ZZ ) ) `  ZZ )
87rneqi 5235 . . . . . 6  |-  ran  (
I eval  (flds  ZZ ) )  =  ran  ( ( I evalSub  (flds  ZZ )
) `  ZZ )
9 simpl 457 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  I  e.  _V )
10 cncrng 18307 . . . . . . . 8  |-fld  e.  CRing
11 zsubrg 18339 . . . . . . . 8  |-  ZZ  e.  (SubRing ` fld )
122subrgcrng 17302 . . . . . . . 8  |-  ( (fld  e. 
CRing  /\  ZZ  e.  (SubRing ` fld ) )  ->  (flds  ZZ )  e.  CRing )
1310, 11, 12mp2an 672 . . . . . . 7  |-  (flds  ZZ )  e.  CRing
1413a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  (flds  ZZ )  e.  CRing )
152subrgring 17301 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1611, 15ax-mp 5 . . . . . . . 8  |-  (flds  ZZ )  e.  Ring
175subrgid 17300 . . . . . . . 8  |-  ( (flds  ZZ )  e.  Ring  ->  ZZ  e.  (SubRing `  (flds  ZZ ) ) )
1816, 17ax-mp 5 . . . . . . 7  |-  ZZ  e.  (SubRing `  (flds  ZZ ) )
1918a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ZZ  e.  (SubRing `  (flds  ZZ )
) )
20 simpr 461 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  f  e.  ZZ )
215, 8, 9, 14, 19, 20mpfconst 18067 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
f } )  e. 
ran  ( I eval  (flds  ZZ )
) )
22 simpl 457 . . . . . 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 461 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  f  e.  I )
265, 8, 22, 23, 24, 25mpfproj 18068 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval  (flds  ZZ ) ) )
27 simp2r 1023 . . . . . 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 1025 . . . . . 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 10885 . . . . . . . 8  |-  ZZ  e.  _V
30 cnfldadd 18293 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
312, 30ressplusg 14613 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  +  =  ( +g  `  (flds  ZZ )
) )
3229, 31ax-mp 5 . . . . . . 7  |-  +  =  ( +g  `  (flds  ZZ ) )
338, 32mpfaddcl 18071 . . . . . 6  |-  ( ( f  e.  ran  (
I eval  (flds  ZZ ) )  /\  g  e.  ran  ( I eval  (flds  ZZ )
) )  ->  (
f  oF  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
3427, 28, 33syl2anc 661 . . . . 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  oF  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
35 cnfldmul 18294 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
362, 35ressmulr 14624 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3729, 36ax-mp 5 . . . . . . 7  |-  x.  =  ( .r `  (flds  ZZ ) )
388, 37mpfmulcl 18072 . . . . . 6  |-  ( ( f  e.  ran  (
I eval  (flds  ZZ ) )  /\  g  e.  ran  ( I eval  (flds  ZZ )
) )  ->  (
f  oF  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
3927, 28, 38syl2anc 661 . . . . 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  oF  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) )
40 eleq1 2539 . . . . 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 2539 . . . . 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 2539 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  f  e.  ran  ( I eval  (flds  ZZ ) ) ) )
43 eleq1 2539 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )
44 eleq1 2539 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( f  oF  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
45 eleq1 2539 . . . . 5  |-  ( b  =  ( f  oF  x.  g )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( f  oF  x.  g )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
46 eleq1 2539 . . . . 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 30612 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval  (flds  ZZ )
) )
48 simprlr 762 . . . . . 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 764 . . . . . 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 30604 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  +  y
)  e.  (mzPoly `  I ) )
5148, 49, 50syl2anc 661 . . . . 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  oF  +  y )  e.  (mzPoly `  I ) )
52 mzpmul 30605 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  x.  y
)  e.  (mzPoly `  I ) )
5348, 49, 52syl2anc 661 . . . . 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  oF  x.  y )  e.  (mzPoly `  I ) )
54 eleq1 2539 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
55 eleq1 2539 . . . . 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 2539 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
57 eleq1 2539 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
58 eleq1 2539 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
59 eleq1 2539 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
60 eleq1 2539 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
61 mzpconst 30601 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
6261adantlr 714 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval  (flds  ZZ ) ) )  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
63 mzpproj 30603 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
6463adantlr 714 . . . . 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 461 . . . . 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 18073 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval  (flds  ZZ ) ) )  -> 
a  e.  (mzPoly `  I ) )
6747, 66impbida 830 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval  (flds  ZZ ) ) ) )
6867eqrdv 2464 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
69 fvprc 5866 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
70 df-evl 18040 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
7170reldmmpt2 6408 . . . . . 6  |-  Rel  dom eval
7271ovprc1 6323 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval  (flds  ZZ ) )  =  (/) )
7372rneqd 5236 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  ran  (/) )
74 rn0 5260 . . . 4  |-  ran  (/)  =  (/)
7573, 74syl6eq 2524 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  (/) )
7669, 75eqtr4d 2511 . 2  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
7768, 76pm2.61i 164 1  |-  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118    C_ wss 3481   (/)c0 3790   {csn 4033    |-> cmpt 4511    X. cxp 5003   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533    ^m cmap 7432   CCcc 9502    + caddc 9507    x. cmul 9509   ZZcz 10876   Basecbs 14506   ↾s cress 14507   +g cplusg 14571   .rcmulr 14572   Ringcrg 17068   CRingccrg 17069  SubRingcsubrg 17294   evalSub ces 18037   eval cevl 18038  ℂfldccnfld 18288  mzPolycmzp 30588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-0g 14713  df-gsum 14714  df-prds 14719  df-pws 14721  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-ghm 16136  df-cntz 16226  df-cmn 16671  df-abl 16672  df-mgp 17012  df-ur 17024  df-srg 17028  df-ring 17070  df-cring 17071  df-rnghom 17234  df-subrg 17296  df-lmod 17383  df-lss 17448  df-lsp 17487  df-assa 17829  df-asp 17830  df-ascl 17831  df-psr 17873  df-mvr 17874  df-mpl 17875  df-evls 18039  df-evl 18040  df-cnfld 18289  df-mzpcl 30589  df-mzp 30590
This theorem is referenced by: (None)
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