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

Theorem fta1b 22438
Description: The assumption that  R be a domain in fta1g 22436 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 17823 . . . 4  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
21simplbi 460 . . 3  |-  ( R  e. IDomn  ->  R  e.  CRing )
31simprbi 464 . . . 4  |-  ( R  e. IDomn  ->  R  e. Domn )
4 domnnzr 17814 . . . 4  |-  ( R  e. Domn  ->  R  e. NzRing )
53, 4syl 16 . . 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 457 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  ->  R  e. IDomn )
13 eldifsn 4158 . . . . . . 7  |-  ( f  e.  ( B  \  {  .0.  } )  <->  ( f  e.  B  /\  f  =/=  .0.  ) )
1413simplbi 460 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  e.  B )
1514adantl 466 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  e.  B )
1613simprbi 464 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  =/=  .0.  )
1716adantl 466 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  =/=  .0.  )
186, 7, 8, 9, 10, 11, 12, 15, 17fta1g 22436 . . . 4  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
( # `  ( `' ( O `  f
) " { W } ) )  <_ 
( D `  f
) )
1918ralrimiva 2881 . . 3  |-  ( R  e. IDomn  ->  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )
202, 5, 193jca 1176 . 2  |-  ( R  e. IDomn  ->  ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) ) )
21 simp1 996 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e.  CRing )
22 simp2 997 . . . 4  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. NzRing )
23 df-ne 2664 . . . . . . . 8  |-  ( x  =/=  W  <->  -.  x  =  W )
24 eqid 2467 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2467 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
26 eqid 2467 . . . . . . . . . 10  |-  (var1 `  R
)  =  (var1 `  R
)
27 eqid 2467 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
28 simpll1 1035 . . . . . . . . . 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 759 . . . . . . . . . 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 760 . . . . . . . . . 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 755 . . . . . . . . . 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 756 . . . . . . . . . 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 1037 . . . . . . . . . . 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 5872 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( O `  f )  =  ( O `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3534cnveqd 5184 . . . . . . . . . . . . . . 15  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  `' ( O `
 f )  =  `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) ) )
3635imaeq1d 5342 . . . . . . . . . . . . . 14  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( `' ( O `  f )
" { W }
)  =  ( `' ( O `  (
x ( .s `  P ) (var1 `  R
) ) ) " { W } ) )
3736fveq2d 5876 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( # `  ( `' ( O `  f ) " { W } ) )  =  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) ) )
38 fveq2 5872 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( D `  f )  =  ( D `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3937, 38breq12d 4466 . . . . . . . . . . . 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 3216 . . . . . . . . . . 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 16 . . . . . . . . . 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 22437 . . . . . . . . 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 615 . . . . . . . 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 218 . . . . . . 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 378 . . . . . 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 434 . . . . 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 2888 . . . 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 17813 . . . 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 664 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. Domn )
5021, 49, 1sylanbrc 664 . 2  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. IDomn )
5120, 50impbii 188 1  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817    \ cdif 3478   {csn 4033   class class class wbr 4453   `'ccnv 5004   "cima 5008   ` cfv 5594  (class class class)co 6295    <_ cle 9641   #chash 12385   Basecbs 14507   .rcmulr 14573   .scvsca 14576   0gc0g 14712   CRingccrg 17071  NzRingcnzr 17775  Domncdomn 17798  IDomncidom 17799  var1cv1 18085  Poly1cpl1 18086  eval1ce1 18221   deg1 cdg1 22320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-ofr 6536  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-subg 16070  df-ghm 16137  df-cntz 16227  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-srg 17030  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-rnghom 17236  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-nzr 17776  df-rlreg 17801  df-domn 17802  df-idom 17803  df-assa 17831  df-asp 17832  df-ascl 17833  df-psr 17875  df-mvr 17876  df-mpl 17877  df-opsr 17879  df-evls 18041  df-evl 18042  df-psr1 18089  df-vr1 18090  df-ply1 18091  df-coe1 18092  df-evl1 18223  df-cnfld 18291  df-mdeg 22321  df-deg1 22322  df-mon1 22399  df-uc1p 22400  df-q1p 22401  df-r1p 22402
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator