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Theorem mzpmfp 29088
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 17894 . . . . . 6  |-  ZZ  =  ( Base ` ring )
2 eqid 2443 . . . . . . . 8  |-  ( I eval ℤring )  =  ( I eval ℤring )
32, 1evlval 17615 . . . . . . 7  |-  ( I eval ℤring )  =  ( ( I evalSub ℤring ) `  ZZ )
43rneqi 5071 . . . . . 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 17890 . . . . . . 7  |-ring  e.  CRing
76a1i 11 . . . . . 6  |-  ( ( I  e.  _V  /\  f  e.  ZZ )  ->ring  e. 
CRing )
8 zringrng 17891 . . . . . . . 8  |-ring  e.  Ring
91subrgid 16872 . . . . . . . 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 17621 . . . . 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 17622 . . . . 5  |-  ( ( I  e.  _V  /\  f  e.  I )  ->  ( g  e.  ( ZZ  ^m  I ) 
|->  ( g `  f
) )  e.  ran  ( I eval ℤring ) )
19 simp2r 1015 . . . . . 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 1017 . . . . . 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 17895 . . . . . . 7  |-  +  =  ( +g  ` ring )
224, 21mpfaddcl 17625 . . . . . 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 17897 . . . . . . 7  |-  x.  =  ( .r ` ring )
254, 24mpfmulcl 17626 . . . . . 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 2503 . . . . 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 2503 . . . . 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 2503 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  f  e.  ran  (
I eval ℤring ) ) )
30 eleq1 2503 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval ℤring ) 
<->  g  e.  ran  (
I eval ℤring ) ) )
31 eleq1 2503 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  +  g )  e.  ran  ( I eval ℤring )
) )
32 eleq1 2503 . . . . 5  |-  ( b  =  ( f  oF  x.  g )  ->  ( b  e. 
ran  ( I eval ℤring )  <->  ( f  oF  x.  g )  e.  ran  ( I eval ℤring )
) )
33 eleq1 2503 . . . . 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 29087 . . . 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 29079 . . . . . 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 29080 . . . . . 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 2503 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
42 eleq1 2503 . . . . 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 2503 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
44 eleq1 2503 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
45 eleq1 2503 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
46 eleq1 2503 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
47 eleq1 2503 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
48 mzpconst 29076 . . . . . 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 29078 . . . . . 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 17627 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval ℤring )
)  ->  a  e.  (mzPoly `  I ) )
5434, 53impbida 828 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval ℤring ) ) )
5554eqrdv 2441 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval ℤring ) )
56 fvprc 5690 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
57 df-evl 17594 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
5857reldmmpt2 6206 . . . . . 6  |-  Rel  dom eval
5958ovprc1 6124 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval ℤring
)  =  (/) )
6059rneqd 5072 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  ran  (/) )
61 rn0 5096 . . . 4  |-  ran  (/)  =  (/)
6260, 61syl6eq 2491 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval ℤring )  =  (/) )
6356, 62eqtr4d 2478 . 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 965    = wceq 1369    e. wcel 1756   _Vcvv 2977   (/)c0 3642   {csn 3882    e. cmpt 4355    X. cxp 4843   ran crn 4846   -->wf 5419   ` cfv 5423  (class class class)co 6096    oFcof 6323    ^m cmap 7219    + caddc 9290    x. cmul 9292   ZZcz 10651   Basecbs 14179   Ringcrg 16650   CRingccrg 16651  SubRingcsubrg 16866   evalSub ces 17591   eval cevl 17592  ℤringzring 17888  mzPolycmzp 29063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-ofr 6326  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-0g 14385  df-gsum 14386  df-prds 14391  df-pws 14393  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-ghm 15750  df-cntz 15840  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-srg 16613  df-rng 16652  df-cring 16653  df-rnghom 16811  df-subrg 16868  df-lmod 16955  df-lss 17019  df-lsp 17058  df-assa 17389  df-asp 17390  df-ascl 17391  df-psr 17428  df-mvr 17429  df-mpl 17430  df-evls 17593  df-evl 17594  df-cnfld 17824  df-zring 17889  df-mzpcl 29064  df-mzp 29065
This theorem is referenced by: (None)
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