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Theorem isdrngo3 26589
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 26588 . 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 3471 . . . . . 6  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
8 difss 3476 . . . . . . . 8  |-  ( X 
\  { Z }
)  C_  X
9 ssrexv 3410 . . . . . . . 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 2611 . . . . . . . . . . . . . . . 16  |-  ( ( y H x )  =  U  ->  (
( y H x )  =/=  Z  <->  U  =/=  Z ) )
1211biimparc 475 . . . . . . . . . . . . . . 15  |-  ( ( U  =/=  Z  /\  ( y H x )  =  U )  ->  ( y H x )  =/=  Z
)
133, 4, 1, 2rngolz 21994 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( Z H x )  =  Z )
14 oveq1 6091 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  Z  ->  (
y H x )  =  ( Z H x ) )
1514eqeq1d 2446 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  Z  ->  (
( y H x )  =  Z  <->  ( Z H x )  =  Z ) )
1613, 15syl5ibrcom 215 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
y  =  Z  -> 
( y H x )  =  Z ) )
1716necon3d 2641 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
( y H x )  =/=  Z  -> 
y  =/=  Z ) )
1817imp 420 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( y H x )  =/=  Z )  ->  y  =/=  Z
)
1912, 18sylan2 462 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( U  =/=  Z  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2019an4s 801 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  ( x  e.  X  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2120anassrs 631 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
y  =/=  Z )
22 pm3.2 436 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
y  =/=  Z  -> 
( y  e.  X  /\  y  =/=  Z
) ) )
2321, 22syl5com 29 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  ( y  e.  X  /\  y  =/=  Z
) ) )
24 eldifsn 3929 . . . . . . . . . . 11  |-  ( y  e.  ( X  \  { Z } )  <->  ( y  e.  X  /\  y  =/=  Z ) )
2523, 24syl6ibr 220 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  y  e.  ( X 
\  { Z }
) ) )
2625imdistanda 676 . . . . . . . . 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 439 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( y H x )  =  U )  <-> 
( ( y H x )  =  U  /\  y  e.  X
) )
28 ancom 439 . . . . . . . . 9  |-  ( ( y  e.  ( X 
\  { Z }
)  /\  ( y H x )  =  U )  <->  ( (
y H x )  =  U  /\  y  e.  ( X  \  { Z } ) ) )
2926, 27, 283imtr4g 263 . . . . . . . 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 2817 . . . . . . 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 197 . . . . . 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 462 . . . . 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 2723 . . . 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 624 . . 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 620 . 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 242 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    \ cdif 3319    C_ wss 3322   {csn 3816   ran crn 4882   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351  GIdcgi 21780   RingOpscrngo 21968   DivRingOpscdrng 21998
This theorem is referenced by:  isfldidl  26692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-riota 6552  df-1o 6727  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-ass 21906  df-exid 21908  df-mgm 21912  df-sgr 21924  df-mndo 21931  df-rngo 21969  df-drngo 21999
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