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Theorem isdrngo3 28933
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 28932 . 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 3589 . . . . . 6  |-  ( x  e.  ( X  \  { Z } )  ->  x  e.  X )
8 difss 3594 . . . . . . . 8  |-  ( X 
\  { Z }
)  C_  X
9 ssrexv 3528 . . . . . . . 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 2733 . . . . . . . . . . . . . . . 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 24060 . . . . . . . . . . . . . . . . . 18  |-  ( ( R  e.  RingOps  /\  x  e.  X )  ->  ( Z H x )  =  Z )
14 oveq1 6210 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  Z  ->  (
y H x )  =  ( Z H x ) )
1514eqeq1d 2456 . . . . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . . . . . 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 822 . . . . . . . . . . . . 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 4111 . . . . . . . . . . 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 2931 . . . . . . 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 2844 . . . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    \ cdif 3436    C_ wss 3439   {csn 3988   ran crn 4952   ` cfv 5529  (class class class)co 6203   1stc1st 6688   2ndc2nd 6689  GIdcgi 23846   RingOpscrngo 24034   DivRingOpscdrng 24064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-om 6590  df-1st 6690  df-2nd 6691  df-1o 7033  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-grpo 23850  df-gid 23851  df-ginv 23852  df-ablo 23941  df-ass 23972  df-exid 23974  df-mgm 23978  df-sgr 23990  df-mndo 23997  df-rngo 24035  df-drngo 24065
This theorem is referenced by:  isfldidl  29036
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