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Theorem fta1b 21653
Description: The assumption that  R be a domain in fta1g 21651 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 17388 . . . 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 17379 . . . 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 4012 . . . . . . 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 21651 . . . 4  |-  ( ( R  e. IDomn  /\  f  e.  ( B  \  {  .0.  } ) )  -> 
( # `  ( `' ( O `  f
) " { W } ) )  <_ 
( D `  f
) )
1918ralrimiva 2811 . . 3  |-  ( R  e. IDomn  ->  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) )
202, 5, 193jca 1168 . 2  |-  ( R  e. IDomn  ->  ( R  e. 
CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `  f ) " { W } ) )  <_ 
( D `  f
) ) )
21 simp1 988 . . 3  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e.  CRing )
22 simp2 989 . . . 4  |-  ( ( R  e.  CRing  /\  R  e. NzRing  /\  A. f  e.  ( B  \  {  .0.  } ) ( # `  ( `' ( O `
 f ) " { W } ) )  <_  ( D `  f ) )  ->  R  e. NzRing )
23 df-ne 2620 . . . . . . . 8  |-  ( x  =/=  W  <->  -.  x  =  W )
24 eqid 2443 . . . . . . . . . 10  |-  ( Base `  R )  =  (
Base `  R )
25 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
26 eqid 2443 . . . . . . . . . 10  |-  (var1 `  R
)  =  (var1 `  R
)
27 eqid 2443 . . . . . . . . . 10  |-  ( .s
`  P )  =  ( .s `  P
)
28 simpll1 1027 . . . . . . . . . 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 1029 . . . . . . . . . . 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 5703 . . . . . . . . . . . . . . . 16  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( O `  f )  =  ( O `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3534cnveqd 5027 . . . . . . . . . . . . . . 15  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  `' ( O `
 f )  =  `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) ) )
3635imaeq1d 5180 . . . . . . . . . . . . . 14  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( `' ( O `  f )
" { W }
)  =  ( `' ( O `  (
x ( .s `  P ) (var1 `  R
) ) ) " { W } ) )
3736fveq2d 5707 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( # `  ( `' ( O `  f ) " { W } ) )  =  ( # `  ( `' ( O `  ( x ( .s
`  P ) (var1 `  R ) ) )
" { W }
) ) )
38 fveq2 5703 . . . . . . . . . . . . 13  |-  ( f  =  ( x ( .s `  P ) (var1 `  R ) )  ->  ( D `  f )  =  ( D `  ( x ( .s `  P
) (var1 `  R ) ) ) )
3937, 38breq12d 4317 . . . . . . . . . . . 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 3082 . . . . . . . . . . 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 21652 . . . . . . . . 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 2820 . . . 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 17378 . . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727    \ cdif 3337   {csn 3889   class class class wbr 4304   `'ccnv 4851   "cima 4855   ` cfv 5430  (class class class)co 6103    <_ cle 9431   #chash 12115   Basecbs 14186   .rcmulr 14251   .scvsca 14254   0gc0g 14390   CRingccrg 16658  NzRingcnzr 17351  Domncdomn 17363  IDomncidom 17364  var1cv1 17644  Poly1cpl1 17645  eval1ce1 17761   deg1 cdg1 21535
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373  ax-mulf 9374
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-iin 4186  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-ofr 6333  df-om 6489  df-1st 6589  df-2nd 6590  df-supp 6703  df-tpos 6757  df-recs 6844  df-rdg 6878  df-1o 6932  df-2o 6933  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-fsupp 7633  df-sup 7703  df-oi 7736  df-card 8121  df-cda 8349  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-fzo 11561  df-seq 11819  df-hash 12116  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-starv 14265  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-unif 14273  df-hom 14274  df-cco 14275  df-0g 14392  df-gsum 14393  df-prds 14398  df-pws 14400  df-mre 14536  df-mrc 14537  df-acs 14539  df-mnd 15427  df-mhm 15476  df-submnd 15477  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mulg 15560  df-subg 15690  df-ghm 15757  df-cntz 15847  df-cmn 16291  df-abl 16292  df-mgp 16604  df-ur 16616  df-srg 16620  df-rng 16659  df-cring 16660  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-rnghom 16818  df-subrg 16875  df-lmod 16962  df-lss 17026  df-lsp 17065  df-nzr 17352  df-rlreg 17366  df-domn 17367  df-idom 17368  df-assa 17396  df-asp 17397  df-ascl 17398  df-psr 17435  df-mvr 17436  df-mpl 17437  df-opsr 17439  df-evls 17600  df-evl 17601  df-psr1 17648  df-vr1 17649  df-ply1 17650  df-coe1 17651  df-evl1 17763  df-cnfld 17831  df-mdeg 21536  df-deg1 21537  df-mon1 21614  df-uc1p 21615  df-q1p 21616  df-r1p 21617
This theorem is referenced by: (None)
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