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Theorem lvolnleat 30394
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 952 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( K  e.  HL  /\  X  e.  V ) )
2 simp3 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  P  e.  A )
3 lvolnleat.l . . . 4  |-  .<_  =  ( le `  K )
4 eqid 2296 . . . 4  |-  ( join `  K )  =  (
join `  K )
5 lvolnleat.a . . . 4  |-  A  =  ( Atoms `  K )
6 lvolnleat.v . . . 4  |-  V  =  ( LVols `  K )
73, 4, 5, 6lvolnle3at 30393 . . 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 1184 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  -.  X  .<_  ( ( P ( join `  K
) P ) (
join `  K ) P ) )
94, 5hlatjidm 30180 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A )  ->  ( P ( join `  K ) P )  =  P )
1093adant2 974 . . . . 5  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( P ( join `  K ) P )  =  P )
1110oveq1d 5889 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( ( P (
join `  K ) P ) ( join `  K ) P )  =  ( P (
join `  K ) P ) )
1211, 10eqtrd 2328 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( ( P (
join `  K ) P ) ( join `  K ) P )  =  P )
1312breq2d 4051 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  P  e.  A )  ->  ( X  .<_  ( ( P ( join `  K
) P ) (
join `  K ) P )  <->  X  .<_  P ) )
148, 13mtbid 291 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   LVolsclvol 30304
This theorem is referenced by:  lvolneatN  30399  lvoln0N  30402  lplncvrlvol  30427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311
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