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Theorem fidomndrng 17357
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 17346 . . . . 5  |-  ( R  e. Domn  ->  R  e.  Ring )
21adantl 463 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  Ring )
3 domnnzr 17345 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
43adantl 463 . . . . . . . . . 10  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e. NzRing )
5 eqid 2441 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2441 . . . . . . . . . . 11  |-  ( 0g
`  R )  =  ( 0g `  R
)
75, 6nzrnz 17320 . . . . . . . . . 10  |-  ( R  e. NzRing  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
84, 7syl 16 . . . . . . . . 9  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( 1r `  R )  =/=  ( 0g `  R ) )
98neneqd 2622 . . . . . . . 8  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  -.  ( 1r `  R
)  =  ( 0g
`  R ) )
10 eqid 2441 . . . . . . . . . 10  |-  (Unit `  R )  =  (Unit `  R )
1110, 6, 50unit 16762 . . . . . . . . 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 3933 . . . . . . 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 16742 . . . . . . 7  |-  (Unit `  R )  C_  B
18 reldisj 3719 . . . . . . 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 2441 . . . . . . . . 9  |-  ( ||r `  R
)  =  ( ||r `  R
)
22 eqid 2441 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
23 simplr 749 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  R  e. Domn )
24 simpll 748 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  B  e.  Fin )
25 simpr 458 . . . . . . . . 9  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x  e.  ( B  \  { ( 0g `  R ) } ) )
26 eqid 2441 . . . . . . . . 9  |-  ( y  e.  B  |->  ( y ( .r `  R
) x ) )  =  ( y  e.  B  |->  ( y ( .r `  R ) x ) )
2716, 6, 5, 21, 22, 23, 24, 25, 26fidomndrnglem 17356 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  R ) ( 1r
`  R ) )
28 eqid 2441 . . . . . . . . . 10  |-  (oppr `  R
)  =  (oppr `  R
)
2928, 16opprbas 16711 . . . . . . . . 9  |-  B  =  ( Base `  (oppr `  R
) )
3028, 6oppr0 16715 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
3128, 5oppr1 16716 . . . . . . . . 9  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
32 eqid 2441 . . . . . . . . 9  |-  ( ||r `  (oppr `  R
) )  =  (
||r `  (oppr
`  R ) )
33 eqid 2441 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
3428opprdomn 17351 . . . . . . . . . 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 2441 . . . . . . . . 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 17356 . . . . . . . 8  |-  ( ( ( B  e.  Fin  /\  R  e. Domn )  /\  x  e.  ( B  \  { ( 0g `  R ) } ) )  ->  x ( ||r `  (oppr
`  R ) ) ( 1r `  R
) )
3810, 5, 21, 28, 32isunit 16739 . . . . . . . 8  |-  ( x  e.  (Unit `  R
)  <->  ( x (
||r `  R ) ( 1r
`  R )  /\  x ( ||r `
 (oppr
`  R ) ) ( 1r `  R
) ) )
3927, 37, 38sylanbrc 659 . . . . . . 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 3359 . . . . 5  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  ( B  \  { ( 0g `  R ) } )  C_  (Unit `  R ) )
4220, 41eqssd 3370 . . . 4  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) )
4316, 10, 6isdrng 16816 . . . 4  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  (Unit `  R )  =  ( B  \  { ( 0g `  R ) } ) ) )
442, 42, 43sylanbrc 659 . . 3  |-  ( ( B  e.  Fin  /\  R  e. Domn )  ->  R  e.  DivRing )
4544ex 434 . 2  |-  ( B  e.  Fin  ->  ( R  e. Domn  ->  R  e.  DivRing ) )
46 drngdomn 17353 . 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 1364    e. wcel 1761    =/= wne 2604    \ cdif 3322    i^i cin 3324    C_ wss 3325   (/)c0 3634   {csn 3874   class class class wbr 4289    e. cmpt 4347   ` cfv 5415  (class class class)co 6090   Fincfn 7306   Basecbs 14170   .rcmulr 14235   0gc0g 14374   1rcur 16593   Ringcrg 16635  opprcoppr 16704   ||rcdsr 16720  Unitcui 16721   DivRingcdr 16812  NzRingcnzr 17317  Domncdomn 17329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-ghm 15738  df-mgp 16582  df-ur 16594  df-rng 16637  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-drng 16814  df-nzr 17318  df-rlreg 17332  df-domn 17333
This theorem is referenced by:  fiidomfld  17358
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