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Theorem mzpmfp 30863
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 18621 . . . . . 6  |-  ZZ  =  ( Base ` ring )
2 eqid 2457 . . . . . . . 8  |-  ( I eval ℤring )  =  ( I eval ℤring )
32, 1evlval 18320 . . . . . . 7  |-  ( I eval ℤring )  =  ( ( I evalSub ℤring ) `  ZZ )
43rneqi 5239 . . . . . 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 18617 . . . . . . 7  |-ring  e.  CRing
76a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->ring  e. 
CRing )
8 zringring 18618 . . . . . . . 8  |-ring  e.  Ring
91subrgid 17558 . . . . . . . 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 18326 . . . . 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 18327 . . . . 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 18622 . . . . . . 7  |-  +  =  ( +g  ` ring )
224, 21mpfaddcl 18330 . . . . . 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 18624 . . . . . . 7  |-  x.  =  ( .r ` ring )
254, 24mpfmulcl 18331 . . . . . 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 2529 . . . . 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 2529 . . . . 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 2529 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  f  e.  ran  (
I eval ℤring ) ) )
30 eleq1 2529 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  g  e.  ran  (
I eval ℤring ) ) )
31 eleq1 2529 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  +  g )  e.  ran  ( I eval ℤring )
) )
32 eleq1 2529 . . . . 5  |-  ( b  =  ( f  oF  x.  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  x.  g )  e.  ran  ( I eval ℤring )
) )
33 eleq1 2529 . . . . 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 30862 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval ℤring ) )
35 simprlr 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 )
) ) )  ->  x  e.  (mzPoly `  I
) )
36 simprrr 766 . . . . . 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 30854 . . . . . 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 30855 . . . . . 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 2529 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
42 eleq1 2529 . . . . 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 2529 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
44 eleq1 2529 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
45 eleq1 2529 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
46 eleq1 2529 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
47 eleq1 2529 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
48 mzpconst 30851 . . . . . 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 30853 . . . . . 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 18332 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval ℤring )
)  ->  a  e.  (mzPoly `  I ) )
5434, 53impbida 832 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval ℤring ) ) )
5554eqrdv 2454 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
56 fvprc 5866 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
57 df-evl 18299 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
5857reldmmpt2 6412 . . . . . 6  |-  Rel  dom eval
5958ovprc1 6327 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval ℤring
)  =  (/) )
6059rneqd 5240 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  ran  (/) )
61 rn0 5264 . . . 4  |-  ran  (/)  =  (/)
6260, 61syl6eq 2514 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  (/) )
6356, 62eqtr4d 2501 . 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 1395    e. wcel 1819   _Vcvv 3109   (/)c0 3793   {csn 4032    |-> cmpt 4515    X. cxp 5006   ran crn 5009   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537    ^m cmap 7438    + caddc 9512    x. cmul 9514   ZZcz 10885   Basecbs 14644   Ringcrg 17325   CRingccrg 17326  SubRingcsubrg 17552   evalSub ces 18296   eval cevl 18297  ℤringzring 18615  mzPolycmzp 30838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-ofr 6540  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-fzo 11822  df-seq 12111  df-hash 12409  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-starv 14727  df-sca 14728  df-vsca 14729  df-ip 14730  df-tset 14731  df-ple 14732  df-ds 14734  df-unif 14735  df-hom 14736  df-cco 14737  df-0g 14859  df-gsum 14860  df-prds 14865  df-pws 14867  df-mre 15003  df-mrc 15004  df-acs 15006  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mhm 16093  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mulg 16187  df-subg 16325  df-ghm 16392  df-cntz 16482  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-srg 17285  df-ring 17327  df-cring 17328  df-rnghom 17491  df-subrg 17554  df-lmod 17641  df-lss 17706  df-lsp 17745  df-assa 18088  df-asp 18089  df-ascl 18090  df-psr 18132  df-mvr 18133  df-mpl 18134  df-evls 18298  df-evl 18299  df-cnfld 18548  df-zring 18616  df-mzpcl 30839  df-mzp 30840
This theorem is referenced by: (None)
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