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Theorem divrngcl 26588
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 26587 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
6 ovres 5987 . . . . 5  |-  ( ( A  e.  ( X 
\  { Z }
)  /\  B  e.  ( X  \  { Z } ) )  -> 
( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  =  ( A H B ) )
76adantl 452 . . . 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 2283 . . . . . . . . 9  |-  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
98grpocl 20867 . . . . . . . 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 1154 . . . . . . 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 452 . . . . . 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 20877 . . . . . . . . . 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 452 . . . . . . . . 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 3303 . . . . . . . . . . . . . . 15  |-  ( X 
\  { Z }
)  C_  X
15 xpss12 4792 . . . . . . . . . . . . . . 15  |-  ( ( ( X  \  { Z } )  C_  X  /\  ( X  \  { Z } )  C_  X
)  ->  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) )  C_  ( X  X.  X ) )
1614, 14, 15mp2an 653 . . . . . . . . . . . . . 14  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  ( X  X.  X )
171, 2, 4rngosm 21048 . . . . . . . . . . . . . . 15  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
18 fdm 5393 . . . . . . . . . . . . . . 15  |-  ( H : ( X  X.  X ) --> X  ->  dom  H  =  ( X  X.  X ) )
1917, 18syl 15 . . . . . . . . . . . . . 14  |-  ( R  e.  RingOps  ->  dom  H  =  ( X  X.  X
) )
2016, 19syl5sseqr 3227 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) )  C_  dom  H )
21 ssdmres 4977 . . . . . . . . . . . . 13  |-  ( ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  dom  H  <->  dom  ( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
2220, 21sylib 188 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )
2322adantr 451 . . . . . . . . . . 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 4881 . . . . . . . . . 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 4898 . . . . . . . . . 10  |-  dom  (
( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( X  \  { Z } )
2624, 25syl6eq 2331 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( X  \  { Z } ) )
2713, 26eqtrd 2315 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( X  \  { Z } ) )
2827eleq2d 2350 . . . . . . 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 2350 . . . . . . 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 691 . . . . . 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 2350 . . . . . 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 258 . . . . 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 418 . . . 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 2358 . . 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 1147 . 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 1218 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   RingOpscrngo 21042   DivRingOpscdrng 21072
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-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-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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-grpo 20858  df-rngo 21043  df-drngo 21073
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