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Theorem isdrngd 16979
Description: Properties that determine a division ring.  I (reciprocal) is normally dependent on  x i.e. read it as  I ( x )." (Contributed by NM, 2-Aug-2013.)
Hypotheses
Ref Expression
isdrngd.b  |-  ( ph  ->  B  =  ( Base `  R ) )
isdrngd.t  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
isdrngd.z  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
isdrngd.u  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
isdrngd.r  |-  ( ph  ->  R  e.  Ring )
isdrngd.n  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =/=  .0.  )
isdrngd.o  |-  ( ph  ->  .1.  =/=  .0.  )
isdrngd.i  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  B )
isdrngd.j  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  =/=  .0.  )
isdrngd.k  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I  .x.  x
)  =  .1.  )
Assertion
Ref Expression
isdrngd  |-  ( ph  ->  R  e.  DivRing )
Distinct variable groups:    x, y,  .0.    x,  .1. , y    x, B, y    y, I    x, R, y    ph, x, y   
x,  .x. , y
Allowed substitution hint:    I( x)

Proof of Theorem isdrngd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 isdrngd.r . . 3  |-  ( ph  ->  R  e.  Ring )
2 difss 3590 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
3 isdrngd.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
42, 3syl5sseq 3511 . . . . 5  |-  ( ph  ->  ( B  \  {  .0.  } )  C_  ( Base `  R ) )
5 eqid 2454 . . . . . 6  |-  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )
6 eqid 2454 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 eqid 2454 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
86, 7mgpbas 16718 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
95, 8ressbas2 14347 . . . . 5  |-  ( ( B  \  {  .0.  } )  C_  ( Base `  R )  ->  ( B  \  {  .0.  }
)  =  ( Base `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
104, 9syl 16 . . . 4  |-  ( ph  ->  ( B  \  {  .0.  } )  =  (
Base `  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) ) )
11 isdrngd.t . . . . 5  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
12 fvex 5808 . . . . . . 7  |-  ( Base `  R )  e.  _V
133, 12syl6eqel 2550 . . . . . 6  |-  ( ph  ->  B  e.  _V )
14 difexg 4547 . . . . . 6  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
15 eqid 2454 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
166, 15mgpplusg 16716 . . . . . . 7  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
175, 16ressplusg 14398 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  (
(mulGrp `  R )s  ( B  \  {  .0.  }
) ) ) )
1813, 14, 173syl 20 . . . . 5  |-  ( ph  ->  ( .r `  R
)  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
1911, 18eqtrd 2495 . . . 4  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
20 eldifsn 4107 . . . . 5  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
21 eldifsn 4107 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
227, 15rngcl 16780 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) )
231, 22syl3an1 1252 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) )  ->  ( x ( .r `  R ) y )  e.  (
Base `  R )
)
24233expib 1191 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) ) )
253eleq2d 2524 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
263eleq2d 2524 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
2725, 26anbi12d 710 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) ) )
2811oveqd 6216 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
2928, 3eleq12d 2536 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .x.  y )  e.  B  <->  ( x ( .r `  R ) y )  e.  ( Base `  R
) ) )
3024, 27, 293imtr4d 268 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B ) )
31303impib 1186 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
32313adant2r 1214 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
33323adant3r 1216 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  e.  B
)
34 isdrngd.n . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =/=  .0.  )
35 eldifsn 4107 . . . . . . 7  |-  ( ( x  .x.  y )  e.  ( B  \  {  .0.  } )  <->  ( (
x  .x.  y )  e.  B  /\  (
x  .x.  y )  =/=  .0.  ) )
3633, 34, 35sylanbrc 664 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  e.  ( B  \  {  .0.  } ) )
3721, 36syl3an3b 1257 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
3820, 37syl3an2b 1256 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
39 eldifi 3585 . . . . . 6  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
40 eldifi 3585 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
41 eldifi 3585 . . . . . 6  |-  ( z  e.  ( B  \  {  .0.  } )  -> 
z  e.  B )
4239, 40, 413anim123i 1173 . . . . 5  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )
437, 15rngass 16783 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  R ) y ) ( .r `  R ) z )  =  ( x ( .r `  R ) ( y ( .r
`  R ) z ) ) )
4443ex 434 . . . . . . . 8  |-  ( R  e.  Ring  ->  ( ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( (
x ( .r `  R ) y ) ( .r `  R
) z )  =  ( x ( .r
`  R ) ( y ( .r `  R ) z ) ) ) )
451, 44syl 16 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) )  ->  ( ( x ( .r `  R
) y ) ( .r `  R ) z )  =  ( x ( .r `  R ) ( y ( .r `  R
) z ) ) ) )
463eleq2d 2524 . . . . . . . 8  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
4725, 26, 463anbi123d 1290 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R ) ) ) )
48 eqidd 2455 . . . . . . . . 9  |-  ( ph  ->  z  =  z )
4911, 28, 48oveq123d 6220 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  y )  .x.  z
)  =  ( ( x ( .r `  R ) y ) ( .r `  R
) z ) )
50 eqidd 2455 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
5111oveqd 6216 . . . . . . . . 9  |-  ( ph  ->  ( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
5211, 50, 51oveq123d 6220 . . . . . . . 8  |-  ( ph  ->  ( x  .x.  (
y  .x.  z )
)  =  ( x ( .r `  R
) ( y ( .r `  R ) z ) ) )
5349, 52eqeq12d 2476 . . . . . . 7  |-  ( ph  ->  ( ( ( x 
.x.  y )  .x.  z )  =  ( x  .x.  ( y 
.x.  z ) )  <-> 
( ( x ( .r `  R ) y ) ( .r
`  R ) z )  =  ( x ( .r `  R
) ( y ( .r `  R ) z ) ) ) )
5445, 47, 533imtr4d 268 . . . . . 6  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
( x  .x.  y
)  .x.  z )  =  ( x  .x.  ( y  .x.  z
) ) ) )
5554imp 429 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
5642, 55sylan2 474 . . . 4  |-  ( (
ph  /\  ( x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) ) )  ->  ( ( x 
.x.  y )  .x.  z )  =  ( x  .x.  ( y 
.x.  z ) ) )
57 eqid 2454 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
587, 57rngidcl 16787 . . . . . . 7  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
591, 58syl 16 . . . . . 6  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
60 isdrngd.u . . . . . 6  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
6159, 60, 33eltr4d 2557 . . . . 5  |-  ( ph  ->  .1.  e.  B )
62 isdrngd.o . . . . 5  |-  ( ph  ->  .1.  =/=  .0.  )
63 eldifsn 4107 . . . . 5  |-  (  .1. 
e.  ( B  \  {  .0.  } )  <->  (  .1.  e.  B  /\  .1.  =/=  .0.  ) )
6461, 62, 63sylanbrc 664 . . . 4  |-  ( ph  ->  .1.  e.  ( B 
\  {  .0.  }
) )
657, 15, 57rnglidm 16790 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
6665ex 434 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( x  e.  ( Base `  R
)  ->  ( ( 1r `  R ) ( .r `  R ) x )  =  x ) )
671, 66syl 16 . . . . . . . 8  |-  ( ph  ->  ( x  e.  (
Base `  R )  ->  ( ( 1r `  R ) ( .r
`  R ) x )  =  x ) )
6811, 60, 50oveq123d 6220 . . . . . . . . 9  |-  ( ph  ->  (  .1.  .x.  x
)  =  ( ( 1r `  R ) ( .r `  R
) x ) )
6968eqeq1d 2456 . . . . . . . 8  |-  ( ph  ->  ( (  .1.  .x.  x )  =  x  <-> 
( ( 1r `  R ) ( .r
`  R ) x )  =  x ) )
7067, 25, 693imtr4d 268 . . . . . . 7  |-  ( ph  ->  ( x  e.  B  ->  (  .1.  .x.  x
)  =  x ) )
7170imp 429 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
7271adantrr 716 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
(  .1.  .x.  x
)  =  x )
7320, 72sylan2b 475 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (  .1.  .x.  x )  =  x )
74 isdrngd.i . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  B )
75 isdrngd.j . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  =/=  .0.  )
76 eldifsn 4107 . . . . . 6  |-  ( I  e.  ( B  \  {  .0.  } )  <->  ( I  e.  B  /\  I  =/= 
.0.  ) )
7774, 75, 76sylanbrc 664 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  ( B  \  {  .0.  } ) )
7820, 77sylan2b 475 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  I  e.  ( B  \  {  .0.  } ) )
79 isdrngd.k . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I  .x.  x
)  =  .1.  )
8020, 79sylan2b 475 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (
I  .x.  x )  =  .1.  )
8110, 19, 38, 56, 64, 73, 78, 80isgrpd 15681 . . 3  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  e. 
Grp )
82 isdrngd.z . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8382sneqd 3996 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  R ) } )
843, 83difeq12d 3582 . . . . . 6  |-  ( ph  ->  ( B  \  {  .0.  } )  =  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
8584oveq2d 6215 . . . . 5  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) ) )
8685eleq1d 2523 . . . 4  |-  ( ph  ->  ( ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  e.  Grp  <->  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  e.  Grp ) )
8786anbi2d 703 . . 3  |-  ( ph  ->  ( ( R  e. 
Ring  /\  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  e.  Grp )  <->  ( R  e.  Ring  /\  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  e.  Grp ) ) )
881, 81, 87mpbi2and 912 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
89 eqid 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
90 eqid 2454 . . 3  |-  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  =  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )
917, 89, 90isdrng2 16964 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
9288, 91sylibr 212 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   _Vcvv 3076    \ cdif 3432    C_ wss 3435   {csn 3984   ` cfv 5525  (class class class)co 6199   Basecbs 14291   ↾s cress 14292   +g cplusg 14356   .rcmulr 14357   0gc0g 14496   Grpcgrp 15528  mulGrpcmgp 16712   1rcur 16724   Ringcrg 16767   DivRingcdr 16954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-tpos 6854  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-nn 10433  df-2 10490  df-3 10491  df-ndx 14294  df-slot 14295  df-base 14296  df-sets 14297  df-ress 14298  df-plusg 14369  df-mulr 14370  df-0g 14498  df-mnd 15533  df-grp 15663  df-minusg 15664  df-mgp 16713  df-ur 16725  df-rng 16769  df-oppr 16837  df-dvdsr 16855  df-unit 16856  df-invr 16886  df-dvr 16897  df-drng 16956
This theorem is referenced by:  isdrngrd  16980  cndrng  17969  erngdvlem4  34958
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