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Theorem divrngcl 26587
Description: The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-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
Assertion
Ref Expression
divrngcl  |-  ( ( R  e.  DivRingOps  /\  A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) )  -> 
( A H B )  e.  ( X 
\  { Z }
) )

Proof of Theorem divrngcl
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
51, 2, 3, 4isdrngo1 26586 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
6 ovres 6216 . . . . 5  |-  ( ( A  e.  ( X 
\  { Z }
)  /\  B  e.  ( X  \  { Z } ) )  -> 
( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  =  ( A H B ) )
76adantl 454 . . . 4  |-  ( ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp )  /\  ( A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) ) )  ->  ( A ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) ) B )  =  ( A H B ) )
8 eqid 2438 . . . . . . . . 9  |-  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
98grpocl 21793 . . . . . . . 8  |-  ( ( ( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp  /\  A  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  /\  B  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )  ->  ( A
( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) ) B )  e.  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) ) )
1093expib 1157 . . . . . . 7  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp  ->  ( ( A  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  /\  B  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) ) )  ->  ( A ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) ) B )  e.  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) ) ) )
1110adantl 454 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( ( A  e.  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  /\  B  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )  ->  ( A
( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) ) B )  e.  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) ) ) )
12 grporndm 21803 . . . . . . . . . 10  |-  ( ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp  ->  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  dom  dom  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) ) )
1312adantl 454 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )
14 difss 3476 . . . . . . . . . . . . . . 15  |-  ( X 
\  { Z }
)  C_  X
15 xpss12 4984 . . . . . . . . . . . . . . 15  |-  ( ( ( X  \  { Z } )  C_  X  /\  ( X  \  { Z } )  C_  X
)  ->  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) )  C_  ( X  X.  X ) )
1614, 14, 15mp2an 655 . . . . . . . . . . . . . 14  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  ( X  X.  X )
171, 2, 4rngosm 21974 . . . . . . . . . . . . . . 15  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
18 fdm 5598 . . . . . . . . . . . . . . 15  |-  ( H : ( X  X.  X ) --> X  ->  dom  H  =  ( X  X.  X ) )
1917, 18syl 16 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  dom  H  =  ( X  X.  X
) )
2016, 19syl5sseqr 3399 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) )  C_  dom  H )
21 ssdmres 5171 . . . . . . . . . . . . 13  |-  ( ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  dom  H  <->  dom  ( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
2220, 21sylib 190 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )
2322adantr 453 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
2423dmeqd 5075 . . . . . . . . . 10  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  dom  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )
25 dmxpid 5092 . . . . . . . . . 10  |-  dom  (
( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( X  \  { Z } )
2624, 25syl6eq 2486 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( X  \  { Z } ) )
2713, 26eqtrd 2470 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( X  \  { Z } ) )
2827eleq2d 2505 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( A  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  <-> 
A  e.  ( X 
\  { Z }
) ) )
2927eleq2d 2505 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( B  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  <-> 
B  e.  ( X 
\  { Z }
) ) )
3028, 29anbi12d 693 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( ( A  e.  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  /\  B  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) )  <->  ( A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) ) ) )
3127eleq2d 2505 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( ( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  e.  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  <->  ( A
( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) ) B )  e.  ( X 
\  { Z }
) ) )
3211, 30, 313imtr3d 260 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ( ( A  e.  ( X 
\  { Z }
)  /\  B  e.  ( X  \  { Z } ) )  -> 
( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  e.  ( X  \  { Z } ) ) )
3332imp 420 . . . 4  |-  ( ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp )  /\  ( A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) ) )  ->  ( A ( H  |`  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) ) ) B )  e.  ( X 
\  { Z }
) )
347, 33eqeltrrd 2513 . . 3  |-  ( ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp )  /\  ( A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) ) )  ->  ( A H B )  e.  ( X  \  { Z } ) )
35343impb 1150 . 2  |-  ( ( ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp )  /\  A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) )  -> 
( A H B )  e.  ( X 
\  { Z }
) )
365, 35syl3an1b 1221 1  |-  ( ( R  e.  DivRingOps  /\  A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) )  -> 
( A H B )  e.  ( X 
\  { Z }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    \ cdif 3319    C_ wss 3322   {csn 3816    X. cxp 4879   dom cdm 4881   ran crn 4882    |` cres 4883   -->wf 5453   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351   GrpOpcgr 21779  GIdcgi 21780   RingOpscrngo 21968   DivRingOpscdrng 21998
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-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-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-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-grpo 21784  df-rngo 21969  df-drngo 21999
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