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Theorem mzpmfp 35660
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 19122 . . . . . 6  |-  ZZ  =  ( Base ` ring )
2 eqid 2471 . . . . . . . 8  |-  ( I eval ℤring )  =  ( I eval ℤring )
32, 1evlval 18824 . . . . . . 7  |-  ( I eval ℤring )  =  ( ( I evalSub ℤring ) `  ZZ )
43rneqi 5067 . . . . . 6  |-  ran  (
I eval ℤring )  =  ran  ( ( I evalSub ℤring ) `  ZZ )
5 simpl 464 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  I  e.  _V )
6 zringcrng 19118 . . . . . . 7  |-ring  e.  CRing
76a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->ring  e. 
CRing )
8 zringring 19119 . . . . . . . 8  |-ring  e.  Ring
91subrgid 18088 . . . . . . . 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 468 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  f  e.  ZZ )
131, 4, 5, 7, 11, 12mpfconst 18830 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
f } )  e. 
ran  ( I eval ℤring ) )
14 simpl 464 . . . . . 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 468 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  f  e.  I )
181, 4, 14, 15, 16, 17mpfproj 18831 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval ℤring ) )
19 simp2r 1057 . . . . . 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 1059 . . . . . 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 19123 . . . . . . 7  |-  +  =  ( +g  ` ring )
224, 21mpfaddcl 18834 . . . . . 6  |-  ( ( f  e.  ran  (
I eval ℤring )  /\  g  e.  ran  ( I eval ℤring ) )  ->  (
f  oF  +  g )  e.  ran  ( I eval ℤring ) )
2319, 20, 22syl2anc 673 . . . . 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 19125 . . . . . . 7  |-  x.  =  ( .r ` ring )
254, 24mpfmulcl 18835 . . . . . 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 673 . . . . 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 2537 . . . . 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 2537 . . . . 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 2537 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  f  e.  ran  (
I eval ℤring ) ) )
30 eleq1 2537 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  g  e.  ran  (
I eval ℤring ) ) )
31 eleq1 2537 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  +  g )  e.  ran  ( I eval ℤring )
) )
32 eleq1 2537 . . . . 5  |-  ( b  =  ( f  oF  x.  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  x.  g )  e.  ran  ( I eval ℤring )
) )
33 eleq1 2537 . . . . 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 35659 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval ℤring ) )
35 simprlr 781 . . . . . 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 783 . . . . . 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 35651 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  +  y
)  e.  (mzPoly `  I ) )
3835, 36, 37syl2anc 673 . . . . 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 35652 . . . . . 6  |-  ( ( x  e.  (mzPoly `  I )  /\  y  e.  (mzPoly `  I )
)  ->  ( x  oF  x.  y
)  e.  (mzPoly `  I ) )
4035, 36, 39syl2anc 673 . . . . 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 2537 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
42 eleq1 2537 . . . . 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 2537 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
44 eleq1 2537 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
45 eleq1 2537 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
46 eleq1 2537 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
47 eleq1 2537 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
48 mzpconst 35648 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  ZZ )  ->  ( ( ZZ  ^m  I )  X.  {
x } )  e.  (mzPoly `  I )
)
4948adantlr 729 . . . . 5  |-  ( ( ( I  e.  _V  /\  a  e.  ran  (
I eval ℤring ) )  /\  x  e.  ZZ )  ->  (
( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) )
50 mzpproj 35650 . . . . . 6  |-  ( ( I  e.  _V  /\  x  e.  I )  ->  ( y  e.  ( ZZ  ^m  I ) 
|->  ( y `  x
) )  e.  (mzPoly `  I ) )
5150adantlr 729 . . . . 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 468 . . . . 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 18836 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval ℤring )
)  ->  a  e.  (mzPoly `  I ) )
5434, 53impbida 850 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval ℤring ) ) )
5554eqrdv 2469 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
56 fvprc 5873 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
57 df-evl 18807 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
5857reldmmpt2 6426 . . . . . 6  |-  Rel  dom eval
5958ovprc1 6339 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval ℤring
)  =  (/) )
6059rneqd 5068 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  ran  (/) )
61 rn0 5092 . . . 4  |-  ran  (/)  =  (/)
6260, 61syl6eq 2521 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  (/) )
6356, 62eqtr4d 2508 . 2  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
6455, 63pm2.61i 169 1  |-  (mzPoly `  I )  =  ran  ( I eval ℤring )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   {csn 3959    |-> cmpt 4454    X. cxp 4837   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308    oFcof 6548    ^m cmap 7490    + caddc 9560    x. cmul 9562   ZZcz 10961   Basecbs 15199   Ringcrg 17858   CRingccrg 17859  SubRingcsubrg 18082   evalSub ces 18804   eval cevl 18805  ℤringzring 19116  mzPolycmzp 35635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-ofr 6551  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-fzo 11943  df-seq 12252  df-hash 12554  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-mulg 16754  df-subg 16892  df-ghm 16959  df-cntz 17049  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-srg 17818  df-ring 17860  df-cring 17861  df-rnghom 18021  df-subrg 18084  df-lmod 18171  df-lss 18234  df-lsp 18273  df-assa 18613  df-asp 18614  df-ascl 18615  df-psr 18657  df-mvr 18658  df-mpl 18659  df-evls 18806  df-evl 18807  df-cnfld 19048  df-zring 19117  df-mzpcl 35636  df-mzp 35637
This theorem is referenced by: (None)
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