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Theorem isdrngo3 30567
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 30566 . 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 3622 . . . . . 6  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
8 difss 3627 . . . . . . . 8  |-  ( X 
\  { Z }
)  C_  X
9 ssrexv 3561 . . . . . . . 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 2738 . . . . . . . . . . . . . . . 16  |-  ( ( y H x )  =  U  ->  (
( y H x )  =/=  Z  <->  U  =/=  Z ) )
1211biimparc 487 . . . . . . . . . . . . . . 15  |-  ( ( U  =/=  Z  /\  ( y H x )  =  U )  ->  ( y H x )  =/=  Z
)
133, 4, 1, 2rngolz 25530 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( Z H x )  =  Z )
14 oveq1 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  Z  ->  (
y H x )  =  ( Z H x ) )
1514eqeq1d 2459 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  Z  ->  (
( y H x )  =  Z  <->  ( Z H x )  =  Z ) )
1613, 15syl5ibrcom 222 . . . . . . . . . . . . . . . . 17  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
y  =  Z  -> 
( y H x )  =  Z ) )
1716necon3d 2681 . . . . . . . . . . . . . . . 16  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  (
( y H x )  =/=  Z  -> 
y  =/=  Z ) )
1817imp 429 . . . . . . . . . . . . . . 15  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( y H x )  =/=  Z )  ->  y  =/=  Z
)
1912, 18sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( R  e.  RingOps  /\  x  e.  X )  /\  ( U  =/=  Z  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2019an4s 826 . . . . . . . . . . . . 13  |-  ( ( ( R  e.  RingOps  /\  U  =/=  Z )  /\  ( x  e.  X  /\  ( y H x )  =  U ) )  ->  y  =/=  Z )
2120anassrs 648 . . . . . . . . . . . 12  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
y  =/=  Z )
22 pm3.2 447 . . . . . . . . . . . 12  |-  ( y  e.  X  ->  (
y  =/=  Z  -> 
( y  e.  X  /\  y  =/=  Z
) ) )
2321, 22syl5com 30 . . . . . . . . . . 11  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  ( y  e.  X  /\  y  =/=  Z
) ) )
24 eldifsn 4157 . . . . . . . . . . 11  |-  ( y  e.  ( X  \  { Z } )  <->  ( y  e.  X  /\  y  =/=  Z ) )
2523, 24syl6ibr 227 . . . . . . . . . 10  |-  ( ( ( ( R  e.  RingOps 
/\  U  =/=  Z
)  /\  x  e.  X )  /\  (
y H x )  =  U )  -> 
( y  e.  X  ->  y  e.  ( X 
\  { Z }
) ) )
2625imdistanda 693 . . . . . . . . 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 450 . . . . . . . . 9  |-  ( ( y  e.  X  /\  ( y H x )  =  U )  <-> 
( ( y H x )  =  U  /\  y  e.  X
) )
28 ancom 450 . . . . . . . . 9  |-  ( ( y  e.  ( X 
\  { Z }
)  /\  ( y H x )  =  U )  <->  ( (
y H x )  =  U  /\  y  e.  ( X  \  { Z } ) ) )
2926, 27, 283imtr4g 270 . . . . . . . 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 2928 . . . . . . 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 204 . . . . . 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 474 . . . . 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 2893 . . . 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 641 . . 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 637 . 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 249 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 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    \ cdif 3468    C_ wss 3471   {csn 4032   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25316   RingOpscrngo 25504   DivRingOpscdrng 25534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-grpo 25320  df-gid 25321  df-ginv 25322  df-ablo 25411  df-ass 25442  df-exid 25444  df-mgmOLD 25448  df-sgrOLD 25460  df-mndo 25467  df-rngo 25505  df-drngo 25535
This theorem is referenced by:  isfldidl  30670
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