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Theorem fta1b 19959
Description: The assumption that  R be a domain in fta1g 19957 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 16291 . . . 4  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
21simplbi 447 . . 3  |-  ( R  e. IDomn  ->  R  e.  CRing )
31simprbi 451 . . . 4  |-  ( R  e. IDomn  ->  R  e. Domn )
4 domnnzr 16282 . . . 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 444 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  ->  R  e. IDomn )
13 eldifsn 3870 . . . . . . 7  |-  ( f  e.  ( B  \  {  .0.  } )  <->  ( f  e.  B  /\  f  =/=  .0.  ) )
1413simplbi 447 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  e.  B )
1514adantl 453 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  e.  B )
1613simprbi 451 . . . . . 6  |-  ( f  e.  ( B  \  {  .0.  } )  -> 
f  =/=  .0.  )
1716adantl 453 . . . . 5  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
f  =/=  .0.  )
186, 7, 8, 9, 10, 11, 12, 15, 17fta1g 19957 . . . 4  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
( # `  ( `' ( O `  f
) " { W } ) )  <_ 
( D `  f
) )
1918ralrimiva 2732 . . 3  |-  ( R  e. IDomn  ->  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )
202, 5, 193jca 1134 . 2  |-  ( R  e. IDomn  ->  ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) ) )
21 simp1 957 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e.  CRing )
22 simp2 958 . . . 4  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. NzRing )
23 df-ne 2552 . . . . . . . 8  |-  ( x  =/=  W  <->  -.  x  =  W )
24 eqid 2387 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2387 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
26 eqid 2387 . . . . . . . . . 10  |-  (var1 `  R
)  =  (var1 `  R
)
27 eqid 2387 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
28 simpll1 996 . . . . . . . . . 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 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
) )  ->  x  e.  ( Base `  R
) )
30 simplrr 738 . . . . . . . . . 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 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 ( .r `  R ) y )  =  W )
32 simprr 734 . . . . . . . . . 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 998 . . . . . . . . . . 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 5668 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( O `  f )  =  ( O `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3534cnveqd 4988 . . . . . . . . . . . . . . 15  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  `' ( O `
 f )  =  `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) ) )
3635imaeq1d 5142 . . . . . . . . . . . . . 14  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( `' ( O `  f )
" { W }
)  =  ( `' ( O `  (
x ( .s `  P ) (var1 `  R
) ) ) " { W } ) )
3736fveq2d 5672 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( # `  ( `' ( O `  f ) " { W } ) )  =  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) ) )
38 fveq2 5668 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( D `  f )  =  ( D `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3937, 38breq12d 4166 . . . . . . . . . . . 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 2992 . . . . . . . . . . 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 19958 . . . . . . . . 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 599 . . . . . . . 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 210 . . . . . . 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 368 . . . . . 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 424 . . . . 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 2741 . . . 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 16281 . . . 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 646 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. Domn )
5021, 49, 1sylanbrc 646 . 2  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. IDomn )
5120, 50impbii 181 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649    \ cdif 3260   {csn 3757   class class class wbr 4153   `'ccnv 4817   "cima 4821   ` cfv 5394  (class class class)co 6020    <_ cle 9054   #chash 11545   Basecbs 13396   .rcmulr 13457   .scvsca 13460   0gc0g 13650   CRingccrg 15588  NzRingcnzr 16255  Domncdomn 16267  IDomncidom 16268  var1cv1 16497  Poly1cpl1 16498  eval1ce1 16500   deg1 cdg1 19844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002  ax-mulf 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-ofr 6245  df-1st 6288  df-2nd 6289  df-tpos 6415  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-ixp 7000  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-cda 7981  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-fz 10976  df-fzo 11066  df-seq 11251  df-hash 11546  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-starv 13471  df-sca 13472  df-vsca 13473  df-tset 13475  df-ple 13476  df-ds 13478  df-unif 13479  df-hom 13480  df-cco 13481  df-prds 13598  df-pws 13600  df-0g 13654  df-gsum 13655  df-mre 13738  df-mrc 13739  df-acs 13741  df-mnd 14617  df-mhm 14665  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mulg 14742  df-subg 14868  df-ghm 14931  df-cntz 15043  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-cring 15591  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-rnghom 15746  df-subrg 15793  df-lmod 15879  df-lss 15936  df-lsp 15975  df-nzr 16256  df-rlreg 16270  df-domn 16271  df-idom 16272  df-assa 16299  df-asp 16300  df-ascl 16301  df-psr 16344  df-mvr 16345  df-mpl 16346  df-evls 16347  df-evl 16348  df-opsr 16352  df-psr1 16503  df-vr1 16504  df-ply1 16505  df-evl1 16507  df-coe1 16508  df-cnfld 16627  df-mdeg 19845  df-deg1 19846  df-mon1 19920  df-uc1p 19921  df-q1p 19922  df-r1p 19923
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