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Theorem fidomndrng 16064
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 16053 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
21adantl 452 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  Ring )
3 domnnzr 16052 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
43adantl 452 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e. NzRing )
5 eqid 2296 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2296 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
75, 6nzrnz 16028 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
84, 7syl 15 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
98neneqd 2475 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 1r `  R
)  =  ( 0g
`  R ) )
10 eqid 2296 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
1110, 6, 50unit 15478 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( ( 0g `  R )  e.  (Unit `  R
)  <->  ( 1r `  R )  =  ( 0g `  R ) ) )
122, 11syl 15 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( ( 0g `  R
)  e.  (Unit `  R )  <->  ( 1r `  R )  =  ( 0g `  R ) ) )
139, 12mtbird 292 . . . . . . 7  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 0g `  R
)  e.  (Unit `  R ) )
14 disjsn 3706 . . . . . . 7  |-  ( ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/)  <->  -.  ( 0g `  R )  e.  (Unit `  R ) )
1513, 14sylibr 203 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/) )
16 fidomndrng.b . . . . . . . 8  |-  B  =  ( Base `  R
)
1716, 10unitss 15458 . . . . . . 7  |-  (Unit `  R )  C_  B
18 reldisj 3511 . . . . . . 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 188 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  C_  ( B  \  { ( 0g
`  R ) } ) )
21 eqid 2296 . . . . . . . . 9  |-  ( ||r `  R
)  =  ( ||r `  R
)
22 eqid 2296 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
23 simplr 731 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  R  e. Domn )
24 simpll 730 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  B  e.  Fin )
25 simpr 447 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  ( B  \  { ( 0g `  R ) } ) )
26 eqid 2296 . . . . . . . . 9  |-  ( y  e.  B  |->  ( y ( .r `  R
) x ) )  =  ( y  e.  B  |->  ( y ( .r `  R ) x ) )
2716, 6, 5, 21, 22, 23, 24, 25, 26fidomndrnglem 16063 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  R ) ( 1r
`  R ) )
28 eqid 2296 . . . . . . . . . 10  |-  (oppr `  R
)  =  (oppr `  R
)
2928, 16opprbas 15427 . . . . . . . . 9  |-  B  =  ( Base `  (oppr `  R
) )
3028, 6oppr0 15431 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
3128, 5oppr1 15432 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
32 eqid 2296 . . . . . . . . 9  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
33 eqid 2296 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3428opprdomn 16058 . . . . . . . . . 10  |-  ( R  e. Domn  ->  (oppr
`  R )  e. Domn
)
3523, 34syl 15 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  (oppr
`  R )  e. Domn
)
36 eqid 2296 . . . . . . . . 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 16063 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
3810, 5, 21, 28, 32isunit 15455 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
3927, 37, 38sylanbrc 645 . . . . . . 7  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  (Unit `  R ) )
4039ex 423 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( x  e.  ( B 
\  { ( 0g
`  R ) } )  ->  x  e.  (Unit `  R ) ) )
4140ssrdv 3198 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( B  \  { ( 0g `  R ) } )  C_  (Unit `  R ) )
4220, 41eqssd 3209 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) )
4316, 10, 6isdrng 15532 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) ) )
442, 42, 43sylanbrc 645 . . 3  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  DivRing )
4544ex 423 . 2  |-  ( B  e.  Fin  ->  ( R  e. Domn  ->  R  e.  DivRing ) )
46 drngdomn 16060 . 2  |-  ( R  e.  DivRing  ->  R  e. Domn )
4745, 46impbid1 194 1  |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Ringcrg 15353   1rcur 15355  opprcoppr 15420   ||rcdsr 15436  Unitcui 15437   DivRingcdr 15528  NzRingcnzr 16025  Domncdomn 16037
This theorem is referenced by:  fiidomfld  16065
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-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
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-iun 3923  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-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-ov 5877  df-oprab 5878  df-mpt2 5879  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-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-nzr 16026  df-rlreg 16040  df-domn 16041
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