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

Theorem fidomndrng 16322
Description: A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
Hypothesis
Ref Expression
fidomndrng.b  |-  B  =  ( Base `  R
)
Assertion
Ref Expression
fidomndrng  |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )

Proof of Theorem fidomndrng
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 domnrng 16311 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
21adantl 453 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  Ring )
3 domnnzr 16310 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
43adantl 453 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e. NzRing )
5 eqid 2404 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2404 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
75, 6nzrnz 16286 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
84, 7syl 16 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
98neneqd 2583 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 1r `  R
)  =  ( 0g
`  R ) )
10 eqid 2404 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
1110, 6, 50unit 15740 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( ( 0g `  R )  e.  (Unit `  R
)  <->  ( 1r `  R )  =  ( 0g `  R ) ) )
122, 11syl 16 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( ( 0g `  R
)  e.  (Unit `  R )  <->  ( 1r `  R )  =  ( 0g `  R ) ) )
139, 12mtbird 293 . . . . . . 7  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 0g `  R
)  e.  (Unit `  R ) )
14 disjsn 3828 . . . . . . 7  |-  ( ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/)  <->  -.  ( 0g `  R )  e.  (Unit `  R ) )
1513, 14sylibr 204 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/) )
16 fidomndrng.b . . . . . . . 8  |-  B  =  ( Base `  R
)
1716, 10unitss 15720 . . . . . . 7  |-  (Unit `  R )  C_  B
18 reldisj 3631 . . . . . . 7  |-  ( (Unit `  R )  C_  B  ->  ( ( (Unit `  R )  i^i  {
( 0g `  R
) } )  =  (/) 
<->  (Unit `  R )  C_  ( B  \  {
( 0g `  R
) } ) ) )
1917, 18ax-mp 8 . . . . . 6  |-  ( ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/)  <->  (Unit `  R )  C_  ( B  \  {
( 0g `  R
) } ) )
2015, 19sylib 189 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  C_  ( B  \  { ( 0g
`  R ) } ) )
21 eqid 2404 . . . . . . . . 9  |-  ( ||r `  R
)  =  ( ||r `  R
)
22 eqid 2404 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
23 simplr 732 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  R  e. Domn )
24 simpll 731 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  B  e.  Fin )
25 simpr 448 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  ( B  \  { ( 0g `  R ) } ) )
26 eqid 2404 . . . . . . . . 9  |-  ( y  e.  B  |->  ( y ( .r `  R
) x ) )  =  ( y  e.  B  |->  ( y ( .r `  R ) x ) )
2716, 6, 5, 21, 22, 23, 24, 25, 26fidomndrnglem 16321 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  R ) ( 1r
`  R ) )
28 eqid 2404 . . . . . . . . . 10  |-  (oppr `  R
)  =  (oppr `  R
)
2928, 16opprbas 15689 . . . . . . . . 9  |-  B  =  ( Base `  (oppr `  R
) )
3028, 6oppr0 15693 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
3128, 5oppr1 15694 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
32 eqid 2404 . . . . . . . . 9  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
33 eqid 2404 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3428opprdomn 16316 . . . . . . . . . 10  |-  ( R  e. Domn  ->  (oppr
`  R )  e. Domn
)
3523, 34syl 16 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  (oppr
`  R )  e. Domn
)
36 eqid 2404 . . . . . . . . 9  |-  ( y  e.  B  |->  ( y ( .r `  (oppr `  R
) ) x ) )  =  ( y  e.  B  |->  ( y ( .r `  (oppr `  R
) ) x ) )
3729, 30, 31, 32, 33, 35, 24, 25, 36fidomndrnglem 16321 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
3810, 5, 21, 28, 32isunit 15717 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
3927, 37, 38sylanbrc 646 . . . . . . 7  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  (Unit `  R ) )
4039ex 424 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( x  e.  ( B 
\  { ( 0g
`  R ) } )  ->  x  e.  (Unit `  R ) ) )
4140ssrdv 3314 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( B  \  { ( 0g `  R ) } )  C_  (Unit `  R ) )
4220, 41eqssd 3325 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) )
4316, 10, 6isdrng 15794 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) ) )
442, 42, 43sylanbrc 646 . . 3  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  DivRing )
4544ex 424 . 2  |-  ( B  e.  Fin  ->  ( R  e. Domn  ->  R  e.  DivRing ) )
46 drngdomn 16318 . 2  |-  ( R  e.  DivRing  ->  R  e. Domn )
4745, 46impbid1 195 1  |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   Fincfn 7068   Basecbs 13424   .rcmulr 13485   0gc0g 13678   Ringcrg 15615   1rcur 15617  opprcoppr 15682   ||rcdsr 15698  Unitcui 15699   DivRingcdr 15790  NzRingcnzr 16283  Domncdomn 16295
This theorem is referenced by:  fiidomfld  16323
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-ghm 14959  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-drng 15792  df-nzr 16284  df-rlreg 16298  df-domn 16299
  Copyright terms: Public domain W3C validator