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Theorem lvoli 29764
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 959 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  Y  e.  B )
2 breq1 4026 . . . 4  |-  ( x  =  X  ->  (
x C Y  <->  X C Y ) )
32rspcev 2884 . . 3  |-  ( ( X  e.  P  /\  X C Y )  ->  E. x  e.  P  x C Y )
433ad2antl3 1119 . 2  |-  ( ( ( K  e.  D  /\  Y  e.  B  /\  X  e.  P
)  /\  X C Y )  ->  E. x  e.  P  x C Y )
5 simpl1 958 . . 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 29762 . . 3  |-  ( K  e.  D  ->  ( Y  e.  V  <->  ( Y  e.  B  /\  E. x  e.  P  x C Y ) ) )
115, 10syl 15 . 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 888 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255   Basecbs 13148    <o ccvr 29452   LPlanesclpl 29681   LVolsclvol 29682
This theorem is referenced by:  lplncvrlvol  29805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-lvols 29689
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