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Theorem isdrngd 17204
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 3631 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
3 isdrngd.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
42, 3syl5sseq 3552 . . . . 5  |-  ( ph  ->  ( B  \  {  .0.  } )  C_  ( Base `  R ) )
5 eqid 2467 . . . . . 6  |-  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R )s  ( B  \  {  .0.  } ) )
6 eqid 2467 . . . . . . 7  |-  (mulGrp `  R )  =  (mulGrp `  R )
7 eqid 2467 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
86, 7mgpbas 16937 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
95, 8ressbas2 14542 . . . . 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 5874 . . . . . . 7  |-  ( Base `  R )  e.  _V
133, 12syl6eqel 2563 . . . . . 6  |-  ( ph  ->  B  e.  _V )
14 difexg 4595 . . . . . 6  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
15 eqid 2467 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
166, 15mgpplusg 16935 . . . . . . 7  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
175, 16ressplusg 14593 . . . . . 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 2508 . . . 4  |-  ( ph  ->  .x.  =  ( +g  `  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) ) ) )
20 eldifsn 4152 . . . . 5  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
21 eldifsn 4152 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
227, 15rngcl 16999 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) )
231, 22syl3an1 1261 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) )  ->  ( x ( .r `  R ) y )  e.  (
Base `  R )
)
24233expib 1199 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) ) )
253eleq2d 2537 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  B  <->  x  e.  ( Base `  R
) ) )
263eleq2d 2537 . . . . . . . . . . . 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 6299 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .x.  y
)  =  ( x ( .r `  R
) y ) )
2928, 3eleq12d 2549 . . . . . . . . . . 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 1194 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
32313adant2r 1223 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
33323adant3r 1225 . . . . . . 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 4152 . . . . . . 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 1266 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
3820, 37syl3an2b 1265 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
39 eldifi 3626 . . . . . 6  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
40 eldifi 3626 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
41 eldifi 3626 . . . . . 6  |-  ( z  e.  ( B  \  {  .0.  } )  -> 
z  e.  B )
4239, 40, 413anim123i 1181 . . . . 5  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )
437, 15rngass 17002 . . . . . . . . 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 2537 . . . . . . . 8  |-  ( ph  ->  ( z  e.  B  <->  z  e.  ( Base `  R
) ) )
4725, 26, 463anbi123d 1299 . . . . . . 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 2468 . . . . . . . . 9  |-  ( ph  ->  z  =  z )
4911, 28, 48oveq123d 6303 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  y )  .x.  z
)  =  ( ( x ( .r `  R ) y ) ( .r `  R
) z ) )
50 eqidd 2468 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
5111oveqd 6299 . . . . . . . . 9  |-  ( ph  ->  ( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
5211, 50, 51oveq123d 6303 . . . . . . . 8  |-  ( ph  ->  ( x  .x.  (
y  .x.  z )
)  =  ( x ( .r `  R
) ( y ( .r `  R ) z ) ) )
5349, 52eqeq12d 2489 . . . . . . 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 2467 . . . . . . . 8  |-  ( 1r
`  R )  =  ( 1r `  R
)
587, 57rngidcl 17006 . . . . . . 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 2570 . . . . 5  |-  ( ph  ->  .1.  e.  B )
62 isdrngd.o . . . . 5  |-  ( ph  ->  .1.  =/=  .0.  )
63 eldifsn 4152 . . . . 5  |-  (  .1. 
e.  ( B  \  {  .0.  } )  <->  (  .1.  e.  B  /\  .1.  =/=  .0.  ) )
6461, 62, 63sylanbrc 664 . . . 4  |-  ( ph  ->  .1.  e.  ( B 
\  {  .0.  }
) )
657, 15, 57rnglidm 17009 . . . . . . . . . 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 6303 . . . . . . . . 9  |-  ( ph  ->  (  .1.  .x.  x
)  =  ( ( 1r `  R ) ( .r `  R
) x ) )
6968eqeq1d 2469 . . . . . . . 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 4152 . . . . . 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 15876 . . 3  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  e. 
Grp )
82 isdrngd.z . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8382sneqd 4039 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  R ) } )
843, 83difeq12d 3623 . . . . . 6  |-  ( ph  ->  ( B  \  {  .0.  } )  =  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
8584oveq2d 6298 . . . . 5  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  =  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) ) )
8685eleq1d 2536 . . . 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 919 . 2  |-  ( ph  ->  ( R  e.  Ring  /\  ( (mulGrp `  R
)s  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )  e.  Grp ) )
89 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
90 eqid 2467 . . 3  |-  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )  =  ( (mulGrp `  R )s  ( ( Base `  R )  \  {
( 0g `  R
) } ) )
917, 89, 90isdrng2 17189 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    C_ wss 3476   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   .rcmulr 14552   0gc0g 14691   Grpcgrp 15723  mulGrpcmgp 16931   1rcur 16943   Ringcrg 16986   DivRingcdr 17179
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-drng 17181
This theorem is referenced by:  isdrngrd  17205  cndrng  18218  erngdvlem4  35787
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