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Theorem divrngcl 26463
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 26462 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
6 ovres 6172 . . . . 5  |-  ( ( A  e.  ( X 
\  { Z }
)  /\  B  e.  ( X  \  { Z } ) )  -> 
( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  =  ( A H B ) )
76adantl 453 . . . 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 2404 . . . . . . . . 9  |-  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
98grpocl 21741 . . . . . . . 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 1156 . . . . . . 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 453 . . . . . 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 21751 . . . . . . . . . 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 453 . . . . . . . . 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 3434 . . . . . . . . . . . . . . 15  |-  ( X 
\  { Z }
)  C_  X
15 xpss12 4940 . . . . . . . . . . . . . . 15  |-  ( ( ( X  \  { Z } )  C_  X  /\  ( X  \  { Z } )  C_  X
)  ->  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) )  C_  ( X  X.  X ) )
1614, 14, 15mp2an 654 . . . . . . . . . . . . . 14  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  ( X  X.  X )
171, 2, 4rngosm 21922 . . . . . . . . . . . . . . 15  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
18 fdm 5554 . . . . . . . . . . . . . . 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 3357 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) )  C_  dom  H )
21 ssdmres 5127 . . . . . . . . . . . . 13  |-  ( ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  dom  H  <->  dom  ( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
2220, 21sylib 189 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )
2322adantr 452 . . . . . . . . . . 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 5031 . . . . . . . . . 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 5048 . . . . . . . . . 10  |-  dom  (
( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( X  \  { Z } )
2624, 25syl6eq 2452 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( X  \  { Z } ) )
2713, 26eqtrd 2436 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( X  \  { Z } ) )
2827eleq2d 2471 . . . . . . 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 2471 . . . . . . 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 692 . . . . . 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 2471 . . . . . 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 259 . . . . 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 419 . . . 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 2479 . . 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 1149 . 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 1220 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    \ cdif 3277    C_ wss 3280   {csn 3774    X. cxp 4835   dom cdm 4837   ran crn 4838    |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   GrpOpcgr 21727  GIdcgi 21728   RingOpscrngo 21916   DivRingOpscdrng 21946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419  df-fv 5421  df-ov 6043  df-1st 6308  df-2nd 6309  df-grpo 21732  df-rngo 21917  df-drngo 21947
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