Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  divrngcl Structured version   Unicode version

Theorem divrngcl 29814
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 29813 . 2  |-  ( R  e.  DivRingOps 
<->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  e. 
GrpOp ) )
6 ovres 6417 . . . . 5  |-  ( ( A  e.  ( X 
\  { Z }
)  /\  B  e.  ( X  \  { Z } ) )  -> 
( A ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) ) B )  =  ( A H B ) )
76adantl 466 . . . 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 2460 . . . . . . . . 9  |-  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ran  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
98grpocl 24728 . . . . . . . 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 1194 . . . . . . 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 466 . . . . . 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 24738 . . . . . . . . . 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 466 . . . . . . . . 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 3624 . . . . . . . . . . . . . . 15  |-  ( X 
\  { Z }
)  C_  X
15 xpss12 5099 . . . . . . . . . . . . . . 15  |-  ( ( ( X  \  { Z } )  C_  X  /\  ( X  \  { Z } )  C_  X
)  ->  ( ( X  \  { Z }
)  X.  ( X 
\  { Z }
) )  C_  ( X  X.  X ) )
1614, 14, 15mp2an 672 . . . . . . . . . . . . . 14  |-  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  ( X  X.  X )
171, 2, 4rngosm 24909 . . . . . . . . . . . . . . 15  |-  ( R  e.  RingOps  ->  H : ( X  X.  X ) --> X )
18 fdm 5726 . . . . . . . . . . . . . . 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 3546 . . . . . . . . . . . . 13  |-  ( R  e.  RingOps  ->  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) )  C_  dom  H )
21 ssdmres 5286 . . . . . . . . . . . . 13  |-  ( ( ( X  \  { Z } )  X.  ( X  \  { Z }
) )  C_  dom  H  <->  dom  ( H  |`  (
( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )
2220, 21sylib 196 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z } ) ) )  =  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )
2322adantr 465 . . . . . . . . . . 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 5196 . . . . . . . . . 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 5213 . . . . . . . . . 10  |-  dom  (
( X  \  { Z } )  X.  ( X  \  { Z }
) )  =  ( X  \  { Z } )
2624, 25syl6eq 2517 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  dom  dom  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
) ) )  =  ( X  \  { Z } ) )
2713, 26eqtrd 2501 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  e. 
GrpOp )  ->  ran  ( H  |`  ( ( X 
\  { Z }
)  X.  ( X 
\  { Z }
) ) )  =  ( X  \  { Z } ) )
2827eleq2d 2530 . . . . . . 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 2530 . . . . . . 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 710 . . . . . 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 2530 . . . . . 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 267 . . . . 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 429 . . . 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 2549 . . 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 1187 . 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 1259 1  |-  ( ( R  e.  DivRingOps  /\  A  e.  ( X  \  { Z } )  /\  B  e.  ( X  \  { Z } ) )  -> 
( A H B )  e.  ( X 
\  { Z }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    \ cdif 3466    C_ wss 3469   {csn 4020    X. cxp 4990   dom cdm 4992   ran crn 4993    |` cres 4994   -->wf 5575   ` cfv 5579  (class class class)co 6275   1stc1st 6772   2ndc2nd 6773   GrpOpcgr 24714  GIdcgi 24715   RingOpscrngo 24903   DivRingOpscdrng 24933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-ov 6278  df-1st 6774  df-2nd 6775  df-grpo 24719  df-rngo 24904  df-drngo 24934
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator