MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fta1b Unicode version

Theorem fta1b 19571
Description: The assumption that  R be a domain in fta1g 19569 is necessary. Here we show that the statement is strong enough to prove that  R is a domain. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
fta1b.p  |-  P  =  (Poly1 `  R )
fta1b.b  |-  B  =  ( Base `  P
)
fta1b.d  |-  D  =  ( deg1  `  R )
fta1b.o  |-  O  =  (eval1 `  R )
fta1b.w  |-  W  =  ( 0g `  R
)
fta1b.z  |-  .0.  =  ( 0g `  P )
Assertion
Ref Expression
fta1b  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) ) )
Distinct variable groups:    B, f    D, f    f, O    R, f    f, W    P, f    .0. , f

Proof of Theorem fta1b
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isidom 16061 . . . 4  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
21simplbi 446 . . 3  |-  ( R  e. IDomn  ->  R  e.  CRing )
31simprbi 450 . . . 4  |-  ( R  e. IDomn  ->  R  e. Domn )
4 domnnzr 16052 . . . 4  |-  ( R  e. Domn  ->  R  e. NzRing )
53, 4syl 15 . . 3  |-  ( R  e. IDomn  ->  R  e. NzRing )
6 fta1b.p . . . . 5  |-  P  =  (Poly1 `  R )
7 fta1b.b . . . . 5  |-  B  =  ( Base `  P
)
8 fta1b.d . . . . 5  |-  D  =  ( deg1  `  R )
9 fta1b.o . . . . 5  |-  O  =  (eval1 `  R )
10 fta1b.w . . . . 5  |-  W  =  ( 0g `  R
)
11 fta1b.z . . . . 5  |-  .0.  =  ( 0g `  P )
12 simpl 443 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  ->  R  e. IDomn )
13 eldifsn 3762 . . . . . . 7  |-  ( f  e.  ( B  \  {  .0.  } )  <->  ( f  e.  B  /\  f  =/=  .0.  ) )
1413simplbi 446 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  e.  B )
1514adantl 452 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  e.  B )
1613simprbi 450 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  =/=  .0.  )
1716adantl 452 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  =/=  .0.  )
186, 7, 8, 9, 10, 11, 12, 15, 17fta1g 19569 . . . 4  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
( # `  ( `' ( O `  f
) " { W } ) )  <_ 
( D `  f
) )
1918ralrimiva 2639 . . 3  |-  ( R  e. IDomn  ->  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )
202, 5, 193jca 1132 . 2  |-  ( R  e. IDomn  ->  ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) ) )
21 simp1 955 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e.  CRing )
22 simp2 956 . . . 4  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. NzRing )
23 df-ne 2461 . . . . . . . 8  |-  ( x  =/=  W  <->  -.  x  =  W )
24 eqid 2296 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2296 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
26 eqid 2296 . . . . . . . . . 10  |-  (var1 `  R
)  =  (var1 `  R
)
27 eqid 2296 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
28 simpll1 994 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  R  e.  CRing )
29 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  x  e.  ( Base `  R
) )
30 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  y  e.  ( Base `  R
) )
31 simprl 732 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  (
x ( .r `  R ) y )  =  W )
32 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  x  =/=  W )
33 simpll3 996 . . . . . . . . . . 11  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )
34 fveq2 5541 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( O `  f )  =  ( O `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3534cnveqd 4873 . . . . . . . . . . . . . . 15  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  `' ( O `
 f )  =  `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) ) )
3635imaeq1d 5027 . . . . . . . . . . . . . 14  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( `' ( O `  f )
" { W }
)  =  ( `' ( O `  (
x ( .s `  P ) (var1 `  R
) ) ) " { W } ) )
3736fveq2d 5545 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( # `  ( `' ( O `  f ) " { W } ) )  =  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) ) )
38 fveq2 5541 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( D `  f )  =  ( D `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3937, 38breq12d 4052 . . . . . . . . . . . 12  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f )  <->  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) )  <_  ( D `  ( x
( .s `  P
) (var1 `  R ) ) ) ) )
4039rspccv 2894 . . . . . . . . . . 11  |-  ( A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
)  ->  ( (
x ( .s `  P ) (var1 `  R
) )  e.  ( B  \  {  .0.  } )  ->  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) )  <_  ( D `  ( x
( .s `  P
) (var1 `  R ) ) ) ) )
4133, 40syl 15 . . . . . . . . . 10  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  (
( x ( .s
`  P ) (var1 `  R ) )  e.  ( B  \  {  .0.  } )  ->  ( # `
 ( `' ( O `  ( x ( .s `  P
) (var1 `  R ) ) ) " { W } ) )  <_ 
( D `  (
x ( .s `  P ) (var1 `  R
) ) ) ) )
426, 7, 8, 9, 10, 11, 24, 25, 26, 27, 28, 29, 30, 31, 32, 41fta1blem 19570 . . . . . . . . 9  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( ( x ( .r `  R ) y )  =  W  /\  x  =/=  W
) )  ->  y  =  W )
4342expr 598 . . . . . . . 8  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( x ( .r
`  R ) y )  =  W )  ->  ( x  =/= 
W  ->  y  =  W ) )
4423, 43syl5bir 209 . . . . . . 7  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( x ( .r
`  R ) y )  =  W )  ->  ( -.  x  =  W  ->  y  =  W ) )
4544orrd 367 . . . . . 6  |-  ( ( ( ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  /\  ( x ( .r
`  R ) y )  =  W )  ->  ( x  =  W  \/  y  =  W ) )
4645ex 423 . . . . 5  |-  ( ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) ) )  ->  ( ( x ( .r `  R
) y )  =  W  ->  ( x  =  W  \/  y  =  W ) ) )
4746ralrimivva 2648 . . . 4  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R ) ( ( x ( .r `  R ) y )  =  W  ->  (
x  =  W  \/  y  =  W )
) )
4824, 25, 10isdomn 16051 . . . 4  |-  ( R  e. Domn 
<->  ( R  e. NzRing  /\  A. x  e.  ( Base `  R ) A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  W  ->  ( x  =  W  \/  y  =  W ) ) ) )
4922, 47, 48sylanbrc 645 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. Domn )
5021, 49, 1sylanbrc 645 . 2  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. IDomn )
5120, 50impbii 180 1  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162   {csn 3653   class class class wbr 4039   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874    <_ cle 8884   #chash 11353   Basecbs 13164   .rcmulr 13225   .scvsca 13228   0gc0g 13416   CRingccrg 15354  NzRingcnzr 16025  Domncdomn 16037  IDomncidom 16038  var1cv1 16267  Poly1cpl1 16268  eval1ce1 16270   deg1 cdg1 19456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-ofr 6095  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-cntz 14809  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-rnghom 15512  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-nzr 16026  df-rlreg 16040  df-domn 16041  df-idom 16042  df-assa 16069  df-asp 16070  df-ascl 16071  df-psr 16114  df-mvr 16115  df-mpl 16116  df-evls 16117  df-evl 16118  df-opsr 16122  df-psr1 16273  df-vr1 16274  df-ply1 16275  df-evl1 16277  df-coe1 16278  df-cnfld 16394  df-mdeg 19457  df-deg1 19458  df-mon1 19532  df-uc1p 19533  df-q1p 19534  df-r1p 19535
  Copyright terms: Public domain W3C validator