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Theorem lvolnlelln 30382
Description: A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lvolnlelln.l  |-  .<_  =  ( le `  K )
lvolnlelln.n  |-  N  =  ( LLines `  K )
lvolnlelln.v  |-  V  =  ( LVols `  K )
Assertion
Ref Expression
lvolnlelln  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  -.  X  .<_  Y )

Proof of Theorem lvolnlelln
Dummy variables  q  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 960 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  Y  e.  N )
2 eqid 2437 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2437 . . . . 5  |-  ( join `  K )  =  (
join `  K )
4 eqid 2437 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 lvolnlelln.n . . . . 5  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln2 30309 . . . 4  |-  ( K  e.  HL  ->  ( Y  e.  N  <->  ( Y  e.  ( Base `  K
)  /\  E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  Y  =  ( p
( join `  K )
q ) ) ) ) )
763ad2ant1 979 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  ( Y  e.  N  <->  ( Y  e.  ( Base `  K )  /\  E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K )
( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) ) ) )
81, 7mpbid 203 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  ( Y  e.  (
Base `  K )  /\  E. p  e.  (
Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  Y  =  ( p (
join `  K )
q ) ) ) )
9 simp11 988 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  K  e.  HL )
10 simp12 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  X  e.  V )
11 simp2l 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  p  e.  ( Atoms `  K )
)
12 simp2r 985 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  q  e.  ( Atoms `  K )
)
13 lvolnlelln.l . . . . . . . 8  |-  .<_  =  ( le `  K )
14 lvolnlelln.v . . . . . . . 8  |-  V  =  ( LVols `  K )
1513, 3, 4, 14lvolnle3at 30380 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  ( p  e.  ( Atoms `  K )  /\  p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) ) )  ->  -.  X  .<_  ( ( p ( join `  K ) p ) ( join `  K
) q ) )
169, 10, 11, 11, 12, 15syl23anc 1192 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  -.  X  .<_  ( ( p (
join `  K )
p ) ( join `  K ) q ) )
17 simp3r 987 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  Y  =  ( p ( join `  K ) q ) )
183, 4hlatjidm 30167 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
199, 11, 18syl2anc 644 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  ( p
( join `  K )
p )  =  p )
2019oveq1d 6097 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  ( (
p ( join `  K
) p ) (
join `  K )
q )  =  ( p ( join `  K
) q ) )
2117, 20eqtr4d 2472 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  Y  =  ( ( p (
join `  K )
p ) ( join `  K ) q ) )
2221breq2d 4225 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  ( X  .<_  Y  <->  X  .<_  ( ( p ( join `  K
) p ) (
join `  K )
q ) ) )
2316, 22mtbird 294 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  /\  ( p  e.  (
Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  /\  ( p  =/=  q  /\  Y  =  (
p ( join `  K
) q ) ) )  ->  -.  X  .<_  Y )
24233exp 1153 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  ( ( p  e.  ( Atoms `  K )  /\  q  e.  ( Atoms `  K ) )  ->  ( ( p  =/=  q  /\  Y  =  ( p (
join `  K )
q ) )  ->  -.  X  .<_  Y ) ) )
2524rexlimdvv 2837 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  ( E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  Y  =  ( p
( join `  K )
q ) )  ->  -.  X  .<_  Y ) )
2625adantld 455 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  ( ( Y  e.  ( Base `  K
)  /\  E. p  e.  ( Atoms `  K ) E. q  e.  ( Atoms `  K ) ( p  =/=  q  /\  Y  =  ( p
( join `  K )
q ) ) )  ->  -.  X  .<_  Y ) )
278, 26mpd 15 1  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  -.  X  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Atomscatm 30062   HLchlt 30149   LLinesclln 30289   LVolsclvol 30291
This theorem is referenced by:  lvolnelln  30387
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297  df-lvols 30298
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