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Theorem divrngcl 26179
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 26178 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
6 ovres 6113 . . . . 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 2366 . . . . . . . . 9  |-  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
98grpocl 21299 . . . . . . . 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 1155 . . . . . . 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 21309 . . . . . . . . . 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 3390 . . . . . . . . . . . . . . 15  |-  ( X 
\  { Z }
)  C_  X
15 xpss12 4895 . . . . . . . . . . . . . . 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 21480 . . . . . . . . . . . . . . 15  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
18 fdm 5499 . . . . . . . . . . . . . . 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 3313 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) )  C_  dom  H )
21 ssdmres 5080 . . . . . . . . . . . . 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 4984 . . . . . . . . . 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 5001 . . . . . . . . . 10  |-  dom  (
( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( X  \  { Z } )
2624, 25syl6eq 2414 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( X  \  { Z } ) )
2713, 26eqtrd 2398 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( X  \  { Z } ) )
2827eleq2d 2433 . . . . . . 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 2433 . . . . . . 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 2433 . . . . . 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 2441 . . 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 1148 . 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 1219 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 935    = wceq 1647    e. wcel 1715    \ cdif 3235    C_ wss 3238   {csn 3729    X. cxp 4790   dom cdm 4792   ran crn 4793    |` cres 4794   -->wf 5354   ` cfv 5358  (class class class)co 5981   1stc1st 6247   2ndc2nd 6248   GrpOpcgr 21285  GIdcgi 21286   RingOpscrngo 21474   DivRingOpscdrng 21504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fo 5364  df-fv 5366  df-ov 5984  df-1st 6249  df-2nd 6250  df-grpo 21290  df-rngo 21475  df-drngo 21505
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