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Theorem isdrngd 15815
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 3434 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
3 isdrngd.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
42, 3syl5sseq 3356 . . . . 5  |-  ( ph  ->  ( B  \  {  .0.  } )  C_  ( Base `  R ) )
5 eqid 2404 . . . . . 6  |-  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )
6 eqid 2404 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 eqid 2404 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
86, 7mgpbas 15609 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
95, 8ressbas2 13475 . . . . 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 5701 . . . . . . 7  |-  ( Base `  R )  e.  _V
133, 12syl6eqel 2492 . . . . . 6  |-  ( ph  ->  B  e.  _V )
14 difexg 4311 . . . . . 6  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
15 eqid 2404 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
166, 15mgpplusg 15607 . . . . . . 7  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
175, 16ressplusg 13526 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  (
(mulGrp `  R )s  ( B  \  {  .0.  }
) ) ) )
1813, 14, 173syl 19 . . . . 5  |-  ( ph  ->  ( .r `  R
)  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
1911, 18eqtrd 2436 . . . 4  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
20 eldifsn 3887 . . . . 5  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
21 eldifsn 3887 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
227, 15rngcl 15632 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) )
231, 22syl3an1 1217 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) )  ->  ( x ( .r `  R ) y )  e.  (
Base `  R )
)
24233expib 1156 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) ) )
253eleq2d 2471 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
263eleq2d 2471 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
2725, 26anbi12d 692 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) ) )
2811oveqd 6057 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
2928, 3eleq12d 2472 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .x.  y )  e.  B  <->  ( x ( .r `  R ) y )  e.  ( Base `  R
) ) )
3024, 27, 293imtr4d 260 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B ) )
31303impib 1151 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
32313adant2r 1179 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
33323adant3r 1181 . . . . . . 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 3887 . . . . . . 7  |-  ( ( x  .x.  y )  e.  ( B  \  {  .0.  } )  <->  ( (
x  .x.  y )  e.  B  /\  (
x  .x.  y )  =/=  .0.  ) )
3633, 34, 35sylanbrc 646 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  e.  ( B  \  {  .0.  } ) )
3721, 36syl3an3b 1222 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
3820, 37syl3an2b 1221 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
39 eldifi 3429 . . . . . 6  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
40 eldifi 3429 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
41 eldifi 3429 . . . . . 6  |-  ( z  e.  ( B  \  {  .0.  } )  -> 
z  e.  B )
4239, 40, 413anim123i 1139 . . . . 5  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )
437, 15rngass 15635 . . . . . . . . 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 424 . . . . . . . 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 2471 . . . . . . . 8  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
4725, 26, 463anbi123d 1254 . . . . . . 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 2405 . . . . . . . . 9  |-  ( ph  ->  z  =  z )
4911, 28, 48oveq123d 6061 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  y )  .x.  z
)  =  ( ( x ( .r `  R ) y ) ( .r `  R
) z ) )
50 eqidd 2405 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
5111oveqd 6057 . . . . . . . . 9  |-  ( ph  ->  ( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
5211, 50, 51oveq123d 6061 . . . . . . . 8  |-  ( ph  ->  ( x  .x.  (
y  .x.  z )
)  =  ( x ( .r `  R
) ( y ( .r `  R ) z ) ) )
5349, 52eqeq12d 2418 . . . . . . 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 260 . . . . . 6  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
( x  .x.  y
)  .x.  z )  =  ( x  .x.  ( y  .x.  z
) ) ) )
5554imp 419 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
5642, 55sylan2 461 . . . 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 2404 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
587, 57rngidcl 15639 . . . . . . 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 2485 . . . . 5  |-  ( ph  ->  .1.  e.  B )
62 isdrngd.o . . . . 5  |-  ( ph  ->  .1.  =/=  .0.  )
63 eldifsn 3887 . . . . 5  |-  (  .1. 
e.  ( B  \  {  .0.  } )  <->  (  .1.  e.  B  /\  .1.  =/=  .0.  ) )
6461, 62, 63sylanbrc 646 . . . 4  |-  ( ph  ->  .1.  e.  ( B 
\  {  .0.  }
) )
657, 15, 57rnglidm 15642 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
6665ex 424 . . . . . . . . 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 6061 . . . . . . . . 9  |-  ( ph  ->  (  .1.  .x.  x
)  =  ( ( 1r `  R ) ( .r `  R
) x ) )
6968eqeq1d 2412 . . . . . . . 8  |-  ( ph  ->  ( (  .1.  .x.  x )  =  x  <-> 
( ( 1r `  R ) ( .r
`  R ) x )  =  x ) )
7067, 25, 693imtr4d 260 . . . . . . 7  |-  ( ph  ->  ( x  e.  B  ->  (  .1.  .x.  x
)  =  x ) )
7170imp 419 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
7271adantrr 698 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
(  .1.  .x.  x
)  =  x )
7320, 72sylan2b 462 . . . 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 3887 . . . . . 6  |-  ( I  e.  ( B  \  {  .0.  } )  <->  ( I  e.  B  /\  I  =/= 
.0.  ) )
7774, 75, 76sylanbrc 646 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  ( B  \  {  .0.  } ) )
7820, 77sylan2b 462 . . . 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 462 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (
I  .x.  x )  =  .1.  )
8110, 19, 38, 56, 64, 73, 78, 80isgrpd 14785 . . 3  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  e. 
Grp )
82 isdrngd.z . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8382sneqd 3787 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  R ) } )
843, 83difeq12d 3426 . . . . . 6  |-  ( ph  ->  ( B  \  {  .0.  } )  =  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
8584oveq2d 6056 . . . . 5  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) ) )
8685eleq1d 2470 . . . 4  |-  ( ph  ->  ( ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  e.  Grp  <->  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  e.  Grp ) )
8786anbi2d 685 . . 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 888 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
89 eqid 2404 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
90 eqid 2404 . . 3  |-  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  =  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )
917, 89, 90isdrng2 15800 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
9288, 91sylibr 204 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    \ cdif 3277    C_ wss 3280   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   .rcmulr 13485   0gc0g 13678   Grpcgrp 14640  mulGrpcmgp 15603   Ringcrg 15615   1rcur 15617   DivRingcdr 15790
This theorem is referenced by:  isdrngrd  15816  cndrng  16685  erngdvlem4  31473
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-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  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-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792
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