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Theorem lvolnleat 30382
Description: An atom cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnleat.l  |-  .<_  =  ( le `  K )
lvolnleat.a  |-  A  =  ( Atoms `  K )
lvolnleat.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvolnleat  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  -.  X  .<_  P )

Proof of Theorem lvolnleat
StepHypRef Expression
1 3simpa 955 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( K  e.  HL  /\  X  e.  V ) )
2 simp3 960 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  P  e.  A )
3 lvolnleat.l . . . 4  |-  .<_  =  ( le `  K )
4 eqid 2438 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 lvolnleat.a . . . 4  |-  A  =  ( Atoms `  K )
6 lvolnleat.v . . . 4  |-  V  =  ( LVols `  K )
73, 4, 5, 6lvolnle3at 30381 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  ( P  e.  A  /\  P  e.  A  /\  P  e.  A ) )  ->  -.  X  .<_  ( ( P ( join `  K
) P ) (
join `  K ) P ) )
81, 2, 2, 2, 7syl13anc 1187 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  -.  X  .<_  ( ( P ( join `  K
) P ) (
join `  K ) P ) )
94, 5hlatjidm 30168 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P ( join `  K ) P )  =  P )
1093adant2 977 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( P ( join `  K ) P )  =  P )
1110oveq1d 6098 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( ( P (
join `  K ) P ) ( join `  K ) P )  =  ( P (
join `  K ) P ) )
1211, 10eqtrd 2470 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( ( P (
join `  K ) P ) ( join `  K ) P )  =  P )
1312breq2d 4226 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( X  .<_  ( ( P ( join `  K
) P ) (
join `  K ) P )  <->  X  .<_  P ) )
148, 13mtbid 293 1  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  -.  X  .<_  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   lecple 13538   joincjn 14403   Atomscatm 30063   HLchlt 30150   LVolsclvol 30292
This theorem is referenced by:  lvolneatN  30387  lvoln0N  30390  lplncvrlvol  30415
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  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-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 29976  df-ol 29978  df-oml 29979  df-covers 30066  df-ats 30067  df-atl 30098  df-cvlat 30122  df-hlat 30151  df-llines 30297  df-lplanes 30298  df-lvols 30299
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