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Theorem isdrngd 17616
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 3617 . . . . . 6  |-  ( B 
\  {  .0.  }
)  C_  B
3 isdrngd.b . . . . . 6  |-  ( ph  ->  B  =  ( Base `  R ) )
42, 3syl5sseq 3537 . . . . 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 17342 . . . . . 6  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
95, 8ressbas2 14774 . . . . 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 5858 . . . . . . 7  |-  ( Base `  R )  e.  _V
133, 12syl6eqel 2550 . . . . . 6  |-  ( ph  ->  B  e.  _V )
14 difexg 4585 . . . . . 6  |-  ( B  e.  _V  ->  ( B  \  {  .0.  }
)  e.  _V )
15 eqid 2454 . . . . . . . 8  |-  ( .r
`  R )  =  ( .r `  R
)
166, 15mgpplusg 17340 . . . . . . 7  |-  ( .r
`  R )  =  ( +g  `  (mulGrp `  R ) )
175, 16ressplusg 14830 . . . . . 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 4141 . . . . 5  |-  ( x  e.  ( B  \  {  .0.  } )  <->  ( x  e.  B  /\  x  =/=  .0.  ) )
21 eldifsn 4141 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  <->  ( y  e.  B  /\  y  =/=  .0.  ) )
227, 15ringcl 17407 . . . . . . . . . . . . 13  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  R
) y )  e.  ( Base `  R
) )
231, 22syl3an1 1259 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) )  ->  ( x ( .r `  R ) y )  e.  (
Base `  R )
)
24233expib 1197 . . . . . . . . . . 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 708 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  e.  B  /\  y  e.  B )  <->  ( x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) ) )
2811oveqd 6287 . . . . . . . . . . . 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 1192 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B  /\  y  e.  B
)  ->  ( x  .x.  y )  e.  B
)
32313adant2r 1221 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  B )  ->  (
x  .x.  y )  e.  B )
33323adant3r 1223 . . . . . . 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 4141 . . . . . . 7  |-  ( ( x  .x.  y )  e.  ( B  \  {  .0.  } )  <->  ( (
x  .x.  y )  e.  B  /\  (
x  .x.  y )  =/=  .0.  ) )
3633, 34, 35sylanbrc 662 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  e.  ( B  \  {  .0.  } ) )
3721, 36syl3an3b 1264 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
3820, 37syl3an2b 1263 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } )  /\  y  e.  ( B  \  {  .0.  } ) )  -> 
( x  .x.  y
)  e.  ( B 
\  {  .0.  }
) )
39 eldifi 3612 . . . . . 6  |-  ( x  e.  ( B  \  {  .0.  } )  ->  x  e.  B )
40 eldifi 3612 . . . . . 6  |-  ( y  e.  ( B  \  {  .0.  } )  -> 
y  e.  B )
41 eldifi 3612 . . . . . 6  |-  ( z  e.  ( B  \  {  .0.  } )  -> 
z  e.  B )
4239, 40, 413anim123i 1179 . . . . 5  |-  ( ( x  e.  ( B 
\  {  .0.  }
)  /\  y  e.  ( B  \  {  .0.  } )  /\  z  e.  ( B  \  {  .0.  } ) )  -> 
( x  e.  B  /\  y  e.  B  /\  z  e.  B
) )
437, 15ringass 17410 . . . . . . . . 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 432 . . . . . . . 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 1297 . . . . . . 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 6291 . . . . . . . 8  |-  ( ph  ->  ( ( x  .x.  y )  .x.  z
)  =  ( ( x ( .r `  R ) y ) ( .r `  R
) z ) )
50 eqidd 2455 . . . . . . . . 9  |-  ( ph  ->  x  =  x )
5111oveqd 6287 . . . . . . . . 9  |-  ( ph  ->  ( y  .x.  z
)  =  ( y ( .r `  R
) z ) )
5211, 50, 51oveq123d 6291 . . . . . . . 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 427 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  -> 
( ( x  .x.  y )  .x.  z
)  =  ( x 
.x.  ( y  .x.  z ) ) )
5642, 55sylan2 472 . . . 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, 57ringidcl 17414 . . . . . . 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 4141 . . . . 5  |-  (  .1. 
e.  ( B  \  {  .0.  } )  <->  (  .1.  e.  B  /\  .1.  =/=  .0.  ) )
6461, 62, 63sylanbrc 662 . . . 4  |-  ( ph  ->  .1.  e.  ( B 
\  {  .0.  }
) )
657, 15, 57ringlidm 17417 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
6665ex 432 . . . . . . . . 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 6291 . . . . . . . . 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 427 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  (  .1.  .x.  x )  =  x )
7271adantrr 714 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
(  .1.  .x.  x
)  =  x )
7320, 72sylan2b 473 . . . 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 4141 . . . . . 6  |-  ( I  e.  ( B  \  {  .0.  } )  <->  ( I  e.  B  /\  I  =/= 
.0.  ) )
7774, 75, 76sylanbrc 662 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  ( B  \  {  .0.  } ) )
7820, 77sylan2b 473 . . . 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 473 . . . 4  |-  ( (
ph  /\  x  e.  ( B  \  {  .0.  } ) )  ->  (
I  .x.  x )  =  .1.  )
8110, 19, 38, 56, 64, 73, 78, 80isgrpd 16274 . . 3  |-  ( ph  ->  ( (mulGrp `  R
)s  ( B  \  {  .0.  } ) )  e. 
Grp )
82 isdrngd.z . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8382sneqd 4028 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  R ) } )
843, 83difeq12d 3609 . . . . . 6  |-  ( ph  ->  ( B  \  {  .0.  } )  =  ( ( Base `  R
)  \  { ( 0g `  R ) } ) )
8584oveq2d 6286 . . . . 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 701 . . 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 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 17601 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14716   ↾s cress 14717   +g cplusg 14784   .rcmulr 14785   0gc0g 14929   Grpcgrp 16252  mulGrpcmgp 17336   1rcur 17348   Ringcrg 17393   DivRingcdr 17591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593
This theorem is referenced by:  isdrngrd  17617  cndrng  18642  erngdvlem4  37114
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