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Theorem isdrngo3 26590
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 26589 . 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 3298 . . . . . 6  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
8 difss 3303 . . . . . . . 8  |-  ( X 
\  { Z }
)  C_  X
9 ssrexv 3238 . . . . . . . 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 8 . . . . . . 7  |-  ( E. y  e.  ( X 
\  { Z }
) ( y H x )  =  U  ->  E. y  e.  X  ( y H x )  =  U )
11 neeq1 2454 . . . . . . . . . . . . . . . 16  |-  ( ( y H x )  =  U  ->  (
( y H x )  =/=  Z  <->  U  =/=  Z ) )
1211biimparc 473 . . . . . . . . . . . . . . 15  |-  ( ( U  =/=  Z  /\  ( y H x )  =  U )  ->  ( y H x )  =/=  Z
)
133, 4, 1, 2rngolz 21068 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( Z H x )  =  Z )
14 oveq1 5865 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  Z  ->  (
y H x )  =  ( Z H x ) )
1514eqeq1d 2291 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  Z  ->  (
( y H x )  =  Z  <->  ( Z H x )  =  Z ) )
1613, 15syl5ibrcom 213 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
y  =  Z  -> 
( y H x )  =  Z ) )
1716necon3d 2484 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
( y H x )  =/=  Z  -> 
y  =/=  Z ) )
1817imp 418 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( y H x )  =/=  Z )  ->  y  =/=  Z
)
1912, 18sylan2 460 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( U  =/=  Z  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2019an4s 799 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  ( x  e.  X  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2120anassrs 629 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
y  =/=  Z )
22 pm3.2 434 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
y  =/=  Z  -> 
( y  e.  X  /\  y  =/=  Z
) ) )
2321, 22syl5com 26 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  ( y  e.  X  /\  y  =/=  Z
) ) )
24 eldifsn 3749 . . . . . . . . . . 11  |-  ( y  e.  ( X  \  { Z } )  <->  ( y  e.  X  /\  y  =/=  Z ) )
2523, 24syl6ibr 218 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  y  e.  ( X 
\  { Z }
) ) )
2625imdistanda 674 . . . . . . . . 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 437 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( y H x )  =  U )  <-> 
( ( y H x )  =  U  /\  y  e.  X
) )
28 ancom 437 . . . . . . . . 9  |-  ( ( y  e.  ( X 
\  { Z }
)  /\  ( y H x )  =  U )  <->  ( (
y H x )  =  U  /\  y  e.  ( X  \  { Z } ) ) )
2926, 27, 283imtr4g 261 . . . . . . . 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 2652 . . . . . . 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 195 . . . . . 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 460 . . . . 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 2559 . . . 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 622 . . 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 618 . 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 240 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   {csn 3640   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072
This theorem is referenced by:  isfldidl  26693
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
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-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-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-drngo 21073
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