Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvoli Structured version   Unicode version

Theorem lvoli 34772
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 1-Jul-2012.)
Hypotheses
Ref Expression
lvolset.b  |-  B  =  ( Base `  K
)
lvolset.c  |-  C  =  (  <o  `  K )
lvolset.p  |-  P  =  ( LPlanes `  K )
lvolset.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvoli  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  V )

Proof of Theorem lvoli
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpl2 1000 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4456 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 3219 . . 3  |-  ( ( X  e.  P  /\  X C Y )  ->  E. x  e.  P  x C Y )
433ad2antl3 1160 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  E. x  e.  P  x C Y )
5 simpl1 999 . . 3  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  K  e.  D )
6 lvolset.b . . . 4  |-  B  =  ( Base `  K
)
7 lvolset.c . . . 4  |-  C  =  (  <o  `  K )
8 lvolset.p . . . 4  |-  P  =  ( LPlanes `  K )
9 lvolset.v . . . 4  |-  V  =  ( LVols `  K )
106, 7, 8, 9islvol 34770 . . 3  |-  ( K  e.  D  ->  ( Y  e.  V  <->  ( Y  e.  B  /\  E. x  e.  P  x C Y ) ) )
115, 10syl 16 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  ( Y  e.  V  <->  ( Y  e.  B  /\  E. x  e.  P  x C Y ) ) )
121, 4, 11mpbir2and 920 1  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   ` cfv 5594   Basecbs 14507    <o ccvr 34460   LPlanesclpl 34689   LVolsclvol 34690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-lvols 34697
This theorem is referenced by:  lplncvrlvol  34813
  Copyright terms: Public domain W3C validator