MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isdrngd Unicode version

Theorem isdrngd 15537
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 3303 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
3 isdrngd.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
42, 3syl5sseq 3226 . . . . 5  |-  ( ph  ->  ( B  \  {  .0.  } )  C_  ( Base `  R ) )
5 eqid 2283 . . . . . 6  |-  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )
6 eqid 2283 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 eqid 2283 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
86, 7mgpbas 15331 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
95, 8ressbas2 13199 . . . . 5  |-  ( ( B  \  {  .0.  } )  C_  ( Base `  R )  ->  ( B  \  {  .0.  }
)  =  ( Base `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
104, 9syl 15 . . . 4  |-  ( ph  ->  ( B  \  {  .0.  } )  =  (
Base `  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) ) ) )
11 isdrngd.t . . . . 5  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
12 fvex 5539 . . . . . . 7  |-  ( Base `  R )  e.  _V
133, 12syl6eqel 2371 . . . . . 6  |-  ( ph  ->  B  e.  _V )
14 difexg 4162 . . . . . 6  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
15 eqid 2283 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
166, 15mgpplusg 15329 . . . . . . 7  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
175, 16ressplusg 13250 . . . . . 6  |-  ( ( B  \  {  .0.  } )  e.  _V  ->  ( .r `  R )  =  ( +g  `  (
(mulGrp `  R )s  ( B  \  {  .0.  }
) ) ) )
1813, 14, 173syl 18 . . . . 5  |-  ( ph  ->  ( .r `  R
)  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
1911, 18eqtrd 2315 . . . 4  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
20 eldifsn 3749 . . . . 5  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
21 eldifsn 3749 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
227, 15rngcl 15354 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) )
231, 22syl3an1 1215 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) )  ->  ( x ( .r `  R ) y )  e.  (
Base `  R )
)
24233expib 1154 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) ) )
253eleq2d 2350 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
263eleq2d 2350 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  R
) ) )
2725, 26anbi12d 691 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) ) )
2811oveqd 5875 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
2928, 3eleq12d 2351 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .x.  y )  e.  B  <->  ( x ( .r `  R ) y )  e.  ( Base `  R
) ) )
3024, 27, 293imtr4d 259 . . . . . . . . . 10  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B ) )
31303impib 1149 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
32313adant2r 1177 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
33323adant3r 1179 . . . . . . 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 3749 . . . . . . 7  |-  ( ( x  .x.  y )  e.  ( B  \  {  .0.  } )  <->  ( (
x  .x.  y )  e.  B  /\  (
x  .x.  y )  =/=  .0.  ) )
3633, 34, 35sylanbrc 645 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  e.  ( B  \  {  .0.  } ) )
3721, 36syl3an3b 1220 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
3820, 37syl3an2b 1219 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
39 eldifi 3298 . . . . . 6  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
40 eldifi 3298 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
41 eldifi 3298 . . . . . 6  |-  ( z  e.  ( B  \  {  .0.  } )  -> 
z  e.  B )
4239, 40, 413anim123i 1137 . . . . 5  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )
437, 15rngass 15357 . . . . . . . . 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 423 . . . . . . . 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 15 . . . . . . 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 2350 . . . . . . . 8  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
4725, 26, 463anbi123d 1252 . . . . . . 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 2284 . . . . . . . . 9  |-  ( ph  ->  z  =  z )
4911, 28, 48oveq123d 5879 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  y )  .x.  z
)  =  ( ( x ( .r `  R ) y ) ( .r `  R
) z ) )
50 eqidd 2284 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
5111oveqd 5875 . . . . . . . . 9  |-  ( ph  ->  ( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
5211, 50, 51oveq123d 5879 . . . . . . . 8  |-  ( ph  ->  ( x  .x.  (
y  .x.  z )
)  =  ( x ( .r `  R
) ( y ( .r `  R ) z ) ) )
5349, 52eqeq12d 2297 . . . . . . 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 259 . . . . . 6  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
( x  .x.  y
)  .x.  z )  =  ( x  .x.  ( y  .x.  z
) ) ) )
5554imp 418 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
5642, 55sylan2 460 . . . 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 2283 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
587, 57rngidcl 15361 . . . . . . 7  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
591, 58syl 15 . . . . . 6  |-  ( ph  ->  ( 1r `  R
)  e.  ( Base `  R ) )
60 isdrngd.u . . . . . . 7  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
6160, 3eleq12d 2351 . . . . . 6  |-  ( ph  ->  (  .1.  e.  B  <->  ( 1r `  R )  e.  ( Base `  R
) ) )
6259, 61mpbird 223 . . . . 5  |-  ( ph  ->  .1.  e.  B )
63 isdrngd.o . . . . 5  |-  ( ph  ->  .1.  =/=  .0.  )
64 eldifsn 3749 . . . . 5  |-  (  .1. 
e.  ( B  \  {  .0.  } )  <->  (  .1.  e.  B  /\  .1.  =/=  .0.  ) )
6562, 63, 64sylanbrc 645 . . . 4  |-  ( ph  ->  .1.  e.  ( B 
\  {  .0.  }
) )
667, 15, 57rnglidm 15364 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
6766ex 423 . . . . . . . . 9  |-  ( R  e.  Ring  ->  ( x  e.  ( Base `  R
)  ->  ( ( 1r `  R ) ( .r `  R ) x )  =  x ) )
681, 67syl 15 . . . . . . . 8  |-  ( ph  ->  ( x  e.  (
Base `  R )  ->  ( ( 1r `  R ) ( .r
`  R ) x )  =  x ) )
6911, 60, 50oveq123d 5879 . . . . . . . . 9  |-  ( ph  ->  (  .1.  .x.  x
)  =  ( ( 1r `  R ) ( .r `  R
) x ) )
7069eqeq1d 2291 . . . . . . . 8  |-  ( ph  ->  ( (  .1.  .x.  x )  =  x  <-> 
( ( 1r `  R ) ( .r
`  R ) x )  =  x ) )
7168, 25, 703imtr4d 259 . . . . . . 7  |-  ( ph  ->  ( x  e.  B  ->  (  .1.  .x.  x
)  =  x ) )
7271imp 418 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
7372adantrr 697 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
(  .1.  .x.  x
)  =  x )
7420, 73sylan2b 461 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (  .1.  .x.  x )  =  x )
75 isdrngd.i . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  B )
76 isdrngd.j . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  =/=  .0.  )
77 eldifsn 3749 . . . . . 6  |-  ( I  e.  ( B  \  {  .0.  } )  <->  ( I  e.  B  /\  I  =/= 
.0.  ) )
7875, 76, 77sylanbrc 645 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  ( B  \  {  .0.  } ) )
7920, 78sylan2b 461 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  I  e.  ( B  \  {  .0.  } ) )
80 isdrngd.k . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I  .x.  x
)  =  .1.  )
8120, 80sylan2b 461 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (
I  .x.  x )  =  .1.  )
8210, 19, 38, 56, 65, 74, 79, 81isgrpd 14507 . . 3  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  e. 
Grp )
83 isdrngd.z . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8483sneqd 3653 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  R ) } )
853, 84difeq12d 3295 . . . . . 6  |-  ( ph  ->  ( B  \  {  .0.  } )  =  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
8685oveq2d 5874 . . . . 5  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) ) )
8786eleq1d 2349 . . . 4  |-  ( ph  ->  ( ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  e.  Grp  <->  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  e.  Grp ) )
8887anbi2d 684 . . 3  |-  ( ph  ->  ( ( R  e. 
Ring  /\  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  e.  Grp )  <->  ( R  e.  Ring  /\  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  e.  Grp ) ) )
891, 82, 88mpbi2and 887 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
90 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
91 eqid 2283 . . 3  |-  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  =  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )
927, 90, 91isdrng2 15522 . 2  |-  ( R  e.  DivRing 
<->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
9389, 92sylibr 203 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362  mulGrpcmgp 15325   Ringcrg 15337   1rcur 15339   DivRingcdr 15512
This theorem is referenced by:  isdrngrd  15538  cndrng  16403  erngdvlem4  31180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514
  Copyright terms: Public domain W3C validator