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Theorem mzpmfpOLD 30864
Description: Relationship between multivariate Z-polynomials and general multivariate polynomial functions. (Contributed by Stefan O'Rear, 20-Mar-2015.) Obsolete version of mzpmfp 30863 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 10893 . . . . . . 7  |-  ZZ  C_  CC
2 eqid 2457 . . . . . . . 8  |-  (flds  ZZ )  =  (flds  ZZ )
3 cnfldbas 18551 . . . . . . . 8  |-  CC  =  ( Base ` fld )
42, 3ressbas2 14702 . . . . . . 7  |-  ( ZZ  C_  CC  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
51, 4ax-mp 5 . . . . . 6  |-  ZZ  =  ( Base `  (flds  ZZ ) )
6 eqid 2457 . . . . . . . 8  |-  ( I eval  (flds  ZZ ) )  =  ( I eval  (flds  ZZ ) )
76, 5evlval 18320 . . . . . . 7  |-  ( I eval  (flds  ZZ ) )  =  ( ( I evalSub  (flds  ZZ ) ) `  ZZ )
87rneqi 5239 . . . . . 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 18566 . . . . . . . 8  |-fld  e.  CRing
11 zsubrg 18598 . . . . . . . 8  |-  ZZ  e.  (SubRing ` fld )
122subrgcrng 17560 . . . . . . . 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 17559 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1611, 15ax-mp 5 . . . . . . . 8  |-  (flds  ZZ )  e.  Ring
175subrgid 17558 . . . . . . . 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 18326 . . . . 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 18327 . . . . 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 10894 . . . . . . . 8  |-  ZZ  e.  _V
30 cnfldadd 18552 . . . . . . . . 9  |-  +  =  ( +g  ` fld )
312, 30ressplusg 14758 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  +  =  ( +g  `  (flds  ZZ )
) )
3229, 31ax-mp 5 . . . . . . 7  |-  +  =  ( +g  `  (flds  ZZ ) )
338, 32mpfaddcl 18330 . . . . . 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 18553 . . . . . . . . 9  |-  x.  =  ( .r ` fld )
362, 35ressmulr 14769 . . . . . . . 8  |-  ( ZZ  e.  _V  ->  x.  =  ( .r `  (flds  ZZ ) ) )
3729, 36ax-mp 5 . . . . . . 7  |-  x.  =  ( .r `  (flds  ZZ ) )
388, 37mpfmulcl 18331 . . . . . 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 2529 . . . . 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 2529 . . . . 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 2529 . . . . 5  |-  ( b  =  f  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  f  e.  ran  ( I eval  (flds  ZZ ) ) ) )
43 eleq1 2529 . . . . 5  |-  ( b  =  g  ->  (
b  e.  ran  (
I eval  (flds  ZZ ) )  <->  g  e.  ran  ( I eval  (flds  ZZ ) ) ) )
44 eleq1 2529 . . . . 5  |-  ( b  =  ( f  oF  +  g )  ->  ( b  e. 
ran  ( I eval  (flds  ZZ )
)  <->  ( f  oF  +  g )  e.  ran  ( I eval  (flds  ZZ ) ) ) )
45 eleq1 2529 . . . . 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 2529 . . . . 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 30862 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  (mzPoly `  I
) )  ->  a  e.  ran  ( I eval  (flds  ZZ )
) )
48 simprlr 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 ) ) ) )  ->  x  e.  (mzPoly `  I )
)
49 simprrr 766 . . . . . 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 30854 . . . . . 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 30855 . . . . . 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 2529 . . . . 5  |-  ( b  =  ( ( ZZ 
^m  I )  X. 
{ x } )  ->  ( b  e.  (mzPoly `  I )  <->  ( ( ZZ  ^m  I
)  X.  { x } )  e.  (mzPoly `  I ) ) )
55 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 )
) )
56 eleq1 2529 . . . . 5  |-  ( b  =  x  ->  (
b  e.  (mzPoly `  I )  <->  x  e.  (mzPoly `  I ) ) )
57 eleq1 2529 . . . . 5  |-  ( b  =  y  ->  (
b  e.  (mzPoly `  I )  <->  y  e.  (mzPoly `  I ) ) )
58 eleq1 2529 . . . . 5  |-  ( b  =  ( x  oF  +  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  +  y )  e.  (mzPoly `  I ) ) )
59 eleq1 2529 . . . . 5  |-  ( b  =  ( x  oF  x.  y )  ->  ( b  e.  (mzPoly `  I )  <->  ( x  oF  x.  y )  e.  (mzPoly `  I ) ) )
60 eleq1 2529 . . . . 5  |-  ( b  =  a  ->  (
b  e.  (mzPoly `  I )  <->  a  e.  (mzPoly `  I ) ) )
61 mzpconst 30851 . . . . . 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 30853 . . . . . 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 18332 . . . 4  |-  ( ( I  e.  _V  /\  a  e.  ran  ( I eval  (flds  ZZ ) ) )  -> 
a  e.  (mzPoly `  I ) )
6747, 66impbida 832 . . 3  |-  ( I  e.  _V  ->  (
a  e.  (mzPoly `  I )  <->  a  e.  ran  ( I eval  (flds  ZZ ) ) ) )
6867eqrdv 2454 . 2  |-  ( I  e.  _V  ->  (mzPoly `  I )  =  ran  ( I eval  (flds  ZZ ) ) )
69 fvprc 5866 . . 3  |-  ( -.  I  e.  _V  ->  (mzPoly `  I )  =  (/) )
70 df-evl 18299 . . . . . . 7  |- eval  =  ( a  e.  _V , 
b  e.  _V  |->  ( ( a evalSub  b ) `
 ( Base `  b
) ) )
7170reldmmpt2 6412 . . . . . 6  |-  Rel  dom eval
7271ovprc1 6327 . . . . 5  |-  ( -.  I  e.  _V  ->  ( I eval  (flds  ZZ ) )  =  (/) )
7372rneqd 5240 . . . 4  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  ran  (/) )
74 rn0 5264 . . . 4  |-  ran  (/)  =  (/)
7573, 74syl6eq 2514 . . 3  |-  ( -.  I  e.  _V  ->  ran  ( I eval  (flds  ZZ ) )  =  (/) )
7669, 75eqtr4d 2501 . 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 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   (/)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   CCcc 9507    + caddc 9512    x. cmul 9514   ZZcz 10885   Basecbs 14644   ↾s cress 14645   +g cplusg 14712   .rcmulr 14713   Ringcrg 17325   CRingccrg 17326  SubRingcsubrg 17552   evalSub ces 18296   eval cevl 18297  ℂfldccnfld 18547  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-mzpcl 30839  df-mzp 30840
This theorem is referenced by: (None)
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