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Theorem mzpmfp 30598
Description: Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) (Revised by AV, 13-Jun-2019.)
Assertion
Ref Expression
mzpmfp  |-  (mzPoly `  I )  =  ran  ( I eval ℤring )

Proof of Theorem mzpmfp
Dummy variables  a 
b  x  y  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zringbas 18364 . . . . . 6  |-  ZZ  =  ( Base ` ring )
2 eqid 2467 . . . . . . . 8  |-  ( I eval ℤring )  =  ( I eval ℤring )
32, 1evlval 18063 . . . . . . 7  |-  ( I eval ℤring )  =  ( ( I evalSub ℤring ) `  ZZ )
43rneqi 5235 . . . . . 6  |-  ran  (
I eval ℤring )  =  ran  ( ( I evalSub ℤring ) `  ZZ )
5 simpl 457 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  I  e.  _V )
6 zringcrng 18360 . . . . . . 7  |-ring  e.  CRing
76a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->ring  e. 
CRing )
8 zringring 18361 . . . . . . . 8  |-ring  e.  Ring
91subrgid 17302 . . . . . . . 8  |-  (ring  e.  Ring  ->  ZZ  e.  (SubRing ` ring ) )
108, 9ax-mp 5 . . . . . . 7  |-  ZZ  e.  (SubRing ` ring )
1110a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ZZ  e.  (SubRing ` ring ) )
12 simpr 461 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  f  e.  ZZ )
131, 4, 5, 7, 11, 12mpfconst 18069 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
f } )  e. 
ran  ( I eval ℤring ) )
14 simpl 457 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  I  e.  _V )
156a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->ring  e. 
CRing )
1610a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ZZ  e.  (SubRing ` ring ) )
17 simpr 461 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  f  e.  I )
181, 4, 14, 15, 16, 17mpfproj 18070 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval ℤring ) )
19 simp2r 1023 . . . . . 6  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval ℤring ) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval ℤring ) ) )  -> 
f  e.  ran  (
I eval ℤring ) )
20 simp3r 1025 . . . . . 6  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval ℤring ) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval ℤring ) ) )  -> 
g  e.  ran  (
I eval ℤring ) )
21 zringplusg 18365 . . . . . . 7  |-  +  =  ( +g  ` ring )
224, 21mpfaddcl 18073 . . . . . 6  |-  ( ( f  e.  ran  (
I eval ℤring )  /\  g  e.  ran  ( I eval ℤring ) )  ->  (
f  oF  +  g )  e.  ran  ( I eval ℤring ) )
2319, 20, 22syl2anc 661 . . . . 5  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval ℤring ) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval ℤring ) ) )  -> 
( f  oF  +  g )  e. 
ran  ( I eval ℤring ) )
24 zringmulr 18367 . . . . . . 7  |-  x.  =  ( .r ` ring )
254, 24mpfmulcl 18074 . . . . . 6  |-  ( ( f  e.  ran  (
I eval ℤring )  /\  g  e.  ran  ( I eval ℤring ) )  ->  (
f  oF  x.  g )  e.  ran  ( I eval ℤring ) )
2619, 20, 25syl2anc 661 . . . . 5  |-  ( ( I  e.  _V  /\  ( f : ( ZZ  ^m  I ) --> ZZ  /\  f  e. 
ran  ( I eval ℤring ) )  /\  (
g : ( ZZ 
^m  I ) --> ZZ 
/\  g  e.  ran  ( I eval ℤring ) ) )  -> 
( f  oF  x.  g )  e. 
ran  ( I eval ℤring ) )
27 eleq1 2539 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ f } )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( ( ZZ 
^m  I )  X. 
{ f } )  e.  ran  ( I eval ℤring )
) )
28 eleq1 2539 . . . . 5  |-  ( b  =  ( g  e.  ( ZZ  ^m  I
)  |->  ( g `  f ) )  -> 
( b  e.  ran  ( I eval ℤring )  <->  ( g  e.  ( ZZ  ^m  I
)  |->  ( g `  f ) )  e. 
ran  ( I eval ℤring ) ) )
29 eleq1 2539 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  f  e.  ran  (
I eval ℤring ) ) )
30 eleq1 2539 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  g  e.  ran  (
I eval ℤring ) ) )
31 eleq1 2539 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  +  g )  e.  ran  ( I eval ℤring )
) )
32 eleq1 2539 . . . . 5  |-  ( b  =  ( f  oF  x.  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  x.  g )  e.  ran  ( I eval ℤring )
) )
33 eleq1 2539 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  a  e.  ran  (
I eval ℤring ) ) )
3413, 18, 23, 26, 27, 28, 29, 30, 31, 32, 33mzpindd 30597 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval ℤring ) )
35 simprlr 762 . . . . . 6  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  (
( x  e.  ran  ( I eval ℤring )  /\  x  e.  (mzPoly `  I )
)  /\  ( y  e.  ran  ( I eval ℤring )  /\  y  e.  (mzPoly `  I )
) ) )  ->  x  e.  (mzPoly `  I
) )
36 simprrr 764 . . . . . 6  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  (
( x  e.  ran  ( I eval ℤring )  /\  x  e.  (mzPoly `  I )
)  /\  ( y  e.  ran  ( I eval ℤring )  /\  y  e.  (mzPoly `  I )
) ) )  -> 
y  e.  (mzPoly `  I ) )
37 mzpadd 30589 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  +  y
)  e.  (mzPoly `  I ) )
3835, 36, 37syl2anc 661 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  (
( x  e.  ran  ( I eval ℤring )  /\  x  e.  (mzPoly `  I )
)  /\  ( y  e.  ran  ( I eval ℤring )  /\  y  e.  (mzPoly `  I )
) ) )  -> 
( x  oF  +  y )  e.  (mzPoly `  I )
)
39 mzpmul 30590 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  x.  y
)  e.  (mzPoly `  I ) )
4035, 36, 39syl2anc 661 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  (
( x  e.  ran  ( I eval ℤring )  /\  x  e.  (mzPoly `  I )
)  /\  ( y  e.  ran  ( I eval ℤring )  /\  y  e.  (mzPoly `  I )
) ) )  -> 
( x  oF  x.  y )  e.  (mzPoly `  I )
)
41 eleq1 2539 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
42 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 )
) )
43 eleq1 2539 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
44 eleq1 2539 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
45 eleq1 2539 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
46 eleq1 2539 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
47 eleq1 2539 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
48 mzpconst 30586 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
4948adantlr 714 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  x  e.  ZZ )  ->  (
( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) )
50 mzpproj 30588 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
5150adantlr 714 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  x  e.  I )  ->  (
y  e.  ( ZZ 
^m  I )  |->  ( y `  x ) )  e.  (mzPoly `  I ) )
52 simpr 461 . . . . 5  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval ℤring )
)  ->  a  e.  ran  ( I eval ℤring ) )
531, 21, 24, 4, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52mpfind 18075 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval ℤring )
)  ->  a  e.  (mzPoly `  I ) )
5434, 53impbida 830 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval ℤring ) ) )
5554eqrdv 2464 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
56 fvprc 5866 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
57 df-evl 18042 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
5857reldmmpt2 6408 . . . . . 6  |-  Rel  dom eval
5958ovprc1 6323 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval ℤring
)  =  (/) )
6059rneqd 5236 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  ran  (/) )
61 rn0 5260 . . . 4  |-  ran  (/)  =  (/)
6260, 61syl6eq 2524 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  (/) )
6356, 62eqtr4d 2511 . 2  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
6455, 63pm2.61i 164 1  |-  (mzPoly `  I )  =  ran  ( I eval ℤring )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3118   (/)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    + caddc 9507    x. cmul 9509   ZZcz 10876   Basecbs 14507   Ringcrg 17070   CRingccrg 17071  SubRingcsubrg 17296   evalSub ces 18039   eval cevl 18040  ℤringzring 18358  mzPolycmzp 30573
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 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-srg 17030  df-ring 17072  df-cring 17073  df-rnghom 17236  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-assa 17831  df-asp 17832  df-ascl 17833  df-psr 17875  df-mvr 17876  df-mpl 17877  df-evls 18041  df-evl 18042  df-cnfld 18291  df-zring 18359  df-mzpcl 30574  df-mzp 30575
This theorem is referenced by: (None)
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