Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isdrngo3 Structured version   Visualization version   Unicode version

Theorem isdrngo3 32210
Description: A division ring is a ring in which  1  =/=  0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
isdivrng1.1  |-  G  =  ( 1st `  R
)
isdivrng1.2  |-  H  =  ( 2nd `  R
)
isdivrng1.3  |-  Z  =  (GId `  G )
isdivrng1.4  |-  X  =  ran  G
isdivrng2.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
isdrngo3  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
Distinct variable groups:    x, H, y    x, X, y    x, Z, y    x, R, y   
x, U, y
Allowed substitution hints:    G( x, y)

Proof of Theorem isdrngo3
StepHypRef Expression
1 isdivrng1.1 . . 3  |-  G  =  ( 1st `  R
)
2 isdivrng1.2 . . 3  |-  H  =  ( 2nd `  R
)
3 isdivrng1.3 . . 3  |-  Z  =  (GId `  G )
4 isdivrng1.4 . . 3  |-  X  =  ran  G
5 isdivrng2.5 . . 3  |-  U  =  (GId `  H )
61, 2, 3, 4, 5isdrngo2 32209 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U ) ) )
7 eldifi 3557 . . . . . 6  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
8 difss 3562 . . . . . . . 8  |-  ( X 
\  { Z }
)  C_  X
9 ssrexv 3496 . . . . . . . 8  |-  ( ( X  \  { Z } )  C_  X  ->  ( E. y  e.  ( X  \  { Z } ) ( y H x )  =  U  ->  E. y  e.  X  ( y H x )  =  U ) )
108, 9ax-mp 5 . . . . . . 7  |-  ( E. y  e.  ( X 
\  { Z }
) ( y H x )  =  U  ->  E. y  e.  X  ( y H x )  =  U )
11 neeq1 2688 . . . . . . . . . . . . . . . 16  |-  ( ( y H x )  =  U  ->  (
( y H x )  =/=  Z  <->  U  =/=  Z ) )
1211biimparc 490 . . . . . . . . . . . . . . 15  |-  ( ( U  =/=  Z  /\  ( y H x )  =  U )  ->  ( y H x )  =/=  Z
)
133, 4, 1, 2rngolz 26141 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( Z H x )  =  Z )
14 oveq1 6302 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  Z  ->  (
y H x )  =  ( Z H x ) )
1514eqeq1d 2455 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  Z  ->  (
( y H x )  =  Z  <->  ( Z H x )  =  Z ) )
1613, 15syl5ibrcom 226 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
y  =  Z  -> 
( y H x )  =  Z ) )
1716necon3d 2647 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
( y H x )  =/=  Z  -> 
y  =/=  Z ) )
1817imp 431 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( y H x )  =/=  Z )  ->  y  =/=  Z
)
1912, 18sylan2 477 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( U  =/=  Z  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2019an4s 836 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  ( x  e.  X  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2120anassrs 654 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
y  =/=  Z )
22 pm3.2 449 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
y  =/=  Z  -> 
( y  e.  X  /\  y  =/=  Z
) ) )
2321, 22syl5com 31 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  ( y  e.  X  /\  y  =/=  Z
) ) )
24 eldifsn 4100 . . . . . . . . . . 11  |-  ( y  e.  ( X  \  { Z } )  <->  ( y  e.  X  /\  y  =/=  Z ) )
2523, 24syl6ibr 231 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  y  e.  ( X 
\  { Z }
) ) )
2625imdistanda 700 . . . . . . . . 9  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  x  e.  X )  ->  ( ( ( y H x )  =  U  /\  y  e.  X )  ->  (
( y H x )  =  U  /\  y  e.  ( X  \  { Z } ) ) ) )
27 ancom 452 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( y H x )  =  U )  <-> 
( ( y H x )  =  U  /\  y  e.  X
) )
28 ancom 452 . . . . . . . . 9  |-  ( ( y  e.  ( X 
\  { Z }
)  /\  ( y H x )  =  U )  <->  ( (
y H x )  =  U  /\  y  e.  ( X  \  { Z } ) ) )
2926, 27, 283imtr4g 274 . . . . . . . 8  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  x  e.  X )  ->  ( ( y  e.  X  /\  ( y H x )  =  U )  ->  (
y  e.  ( X 
\  { Z }
)  /\  ( y H x )  =  U ) ) )
3029reximdv2 2860 . . . . . . 7  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  x  e.  X )  ->  ( E. y  e.  X  ( y H x )  =  U  ->  E. y  e.  ( X  \  { Z } ) ( y H x )  =  U ) )
3110, 30impbid2 208 . . . . . 6  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  x  e.  X )  ->  ( E. y  e.  ( X  \  { Z } ) ( y H x )  =  U  <->  E. y  e.  X  ( y H x )  =  U ) )
327, 31sylan2 477 . . . . 5  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  x  e.  ( X  \  { Z } ) )  ->  ( E. y  e.  ( X  \  { Z } ) ( y H x )  =  U  <->  E. y  e.  X  ( y H x )  =  U ) )
3332ralbidva 2826 . . . 4  |-  ( ( R  e.  RingOps  /\  U  =/=  Z )  ->  ( A. x  e.  ( X  \  { Z }
) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U  <->  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) )
3433pm5.32da 647 . . 3  |-  ( R  e.  RingOps  ->  ( ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U )  <->  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
3534pm5.32i 643 . 2  |-  ( ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U ) )  <->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
366, 35bitri 253 1  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z } ) E. y  e.  X  ( y H x )  =  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1446    e. wcel 1889    =/= wne 2624   A.wral 2739   E.wrex 2740    \ cdif 3403    C_ wss 3406   {csn 3970   ran crn 4838   ` cfv 5585  (class class class)co 6295   1stc1st 6796   2ndc2nd 6797  GIdcgi 25927   RingOpscrngo 26115   DivRingOpscdrng 26145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  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-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-om 6698  df-1st 6798  df-2nd 6799  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-grpo 25931  df-gid 25932  df-ginv 25933  df-ablo 26022  df-ass 26053  df-exid 26055  df-mgmOLD 26059  df-sgrOLD 26071  df-mndo 26078  df-rngo 26116  df-drngo 26146
This theorem is referenced by:  isfldidl  32313
  Copyright terms: Public domain W3C validator