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Theorem lvoli 30434
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 962 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4217 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 3054 . . 3  |-  ( ( X  e.  P  /\  X C Y )  ->  E. x  e.  P  x C Y )
433ad2antl3 1122 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  E. x  e.  P  x C Y )
5 simpl1 961 . . 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 30432 . . 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 890 1  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   ` cfv 5456   Basecbs 13471    <o ccvr 30122   LPlanesclpl 30351   LVolsclvol 30352
This theorem is referenced by:  lplncvrlvol  30475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-lvols 30359
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