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Theorem fidomndrng 17826
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 domnring 17815 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
21adantl 466 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  Ring )
3 domnnzr 17814 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
43adantl 466 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e. NzRing )
5 eqid 2467 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2467 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
75, 6nzrnz 17778 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
84, 7syl 16 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
98neneqd 2669 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 1r `  R
)  =  ( 0g
`  R ) )
10 eqid 2467 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
1110, 6, 50unit 17201 . . . . . . . . 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 301 . . . . . . 7  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 0g `  R
)  e.  (Unit `  R ) )
14 disjsn 4094 . . . . . . 7  |-  ( ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/)  <->  -.  ( 0g `  R )  e.  (Unit `  R ) )
1513, 14sylibr 212 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/) )
16 fidomndrng.b . . . . . . . 8  |-  B  =  ( Base `  R
)
1716, 10unitss 17181 . . . . . . 7  |-  (Unit `  R )  C_  B
18 reldisj 3875 . . . . . . 7  |-  ( (Unit `  R )  C_  B  ->  ( ( (Unit `  R )  i^i  {
( 0g `  R
) } )  =  (/) 
<->  (Unit `  R )  C_  ( B  \  {
( 0g `  R
) } ) ) )
1917, 18ax-mp 5 . . . . . 6  |-  ( ( (Unit `  R )  i^i  { ( 0g `  R ) } )  =  (/)  <->  (Unit `  R )  C_  ( B  \  {
( 0g `  R
) } ) )
2015, 19sylib 196 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  C_  ( B  \  { ( 0g
`  R ) } ) )
21 eqid 2467 . . . . . . . . 9  |-  ( ||r `  R
)  =  ( ||r `  R
)
22 eqid 2467 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
23 simplr 754 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  R  e. Domn )
24 simpll 753 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  B  e.  Fin )
25 simpr 461 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  ( B  \  { ( 0g `  R ) } ) )
26 eqid 2467 . . . . . . . . 9  |-  ( y  e.  B  |->  ( y ( .r `  R
) x ) )  =  ( y  e.  B  |->  ( y ( .r `  R ) x ) )
2716, 6, 5, 21, 22, 23, 24, 25, 26fidomndrnglem 17825 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  R ) ( 1r
`  R ) )
28 eqid 2467 . . . . . . . . . 10  |-  (oppr `  R
)  =  (oppr `  R
)
2928, 16opprbas 17150 . . . . . . . . 9  |-  B  =  ( Base `  (oppr `  R
) )
3028, 6oppr0 17154 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
3128, 5oppr1 17155 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
32 eqid 2467 . . . . . . . . 9  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
33 eqid 2467 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3428opprdomn 17820 . . . . . . . . . 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 2467 . . . . . . . . 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 17825 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
3810, 5, 21, 28, 32isunit 17178 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
3927, 37, 38sylanbrc 664 . . . . . . 7  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  (Unit `  R ) )
4039ex 434 . . . . . 6  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( x  e.  ( B 
\  { ( 0g
`  R ) } )  ->  x  e.  (Unit `  R ) ) )
4140ssrdv 3515 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( B  \  { ( 0g `  R ) } )  C_  (Unit `  R ) )
4220, 41eqssd 3526 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) )
4316, 10, 6isdrng 17271 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) ) )
442, 42, 43sylanbrc 664 . . 3  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  DivRing )
4544ex 434 . 2  |-  ( B  e.  Fin  ->  ( R  e. Domn  ->  R  e.  DivRing ) )
46 drngdomn 17822 . 2  |-  ( R  e.  DivRing  ->  R  e. Domn )
4745, 46impbid1 203 1  |-  ( B  e.  Fin  ->  ( R  e. Domn  <->  R  e.  DivRing ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3478    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453    |-> cmpt 4511   ` cfv 5594  (class class class)co 6295   Fincfn 7528   Basecbs 14507   .rcmulr 14573   0gc0g 14712   1rcur 17025   Ringcrg 17070  opprcoppr 17143   ||rcdsr 17159  Unitcui 17160   DivRingcdr 17267  NzRingcnzr 17775  Domncdomn 17798
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-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
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-iun 4333  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-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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  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-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-ghm 16137  df-mgp 17014  df-ur 17026  df-ring 17072  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-drng 17269  df-nzr 17776  df-rlreg 17801  df-domn 17802
This theorem is referenced by:  fiidomfld  17827
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