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

Proof of Theorem lvolnlelpln
Dummy variables  r 
q  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 959 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  Y  e.  P )
2 eqid 2436 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 lvolnlelpln.l . . . . 5  |-  .<_  =  ( le `  K )
4 eqid 2436 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2436 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 lvolnlelpln.p . . . . 5  |-  P  =  ( LPlanes `  K )
72, 3, 4, 5, 6islpln2 30333 . . . 4  |-  ( K  e.  HL  ->  ( Y  e.  P  <->  ( Y  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q (
join `  K )
r )  /\  Y  =  ( ( q ( join `  K
) r ) (
join `  K )
s ) ) ) ) )
873ad2ant1 978 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  ( Y  e.  P  <->  ( Y  e.  ( Base `  K )  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K
) r )  /\  Y  =  ( (
q ( join `  K
) r ) (
join `  K )
s ) ) ) ) )
91, 8mpbid 202 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  ( Y  e.  (
Base `  K )  /\  E. q  e.  (
Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K )
( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K
) r )  /\  Y  =  ( (
q ( join `  K
) r ) (
join `  K )
s ) ) ) )
10 simp1l1 1050 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  K  e.  HL )
11 simp1l2 1051 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  X  e.  V )
12 simp1r 982 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  q  e.  ( Atoms `  K )
)
13 simp2l 983 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  r  e.  ( Atoms `  K )
)
14 simp2r 984 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  s  e.  ( Atoms `  K )
)
15 lvolnlelpln.v . . . . . . . . 9  |-  V  =  ( LVols `  K )
163, 4, 5, 15lvolnle3at 30379 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  X  e.  V )  /\  ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) ) )  ->  -.  X  .<_  ( ( q ( join `  K ) r ) ( join `  K
) s ) )
1710, 11, 12, 13, 14, 16syl23anc 1191 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  -.  X  .<_  ( ( q (
join `  K )
r ) ( join `  K ) s ) )
18 simp33 995 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  Y  =  ( ( q (
join `  K )
r ) ( join `  K ) s ) )
1918breq2d 4224 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  ( X  .<_  Y  <->  X  .<_  ( ( q ( join `  K
) r ) (
join `  K )
s ) ) )
2017, 19mtbird 293 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K )
)  /\  ( r  e.  ( Atoms `  K )  /\  s  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K ) r )  /\  Y  =  ( ( q ( join `  K ) r ) ( join `  K
) s ) ) )  ->  -.  X  .<_  Y )
21203exp 1152 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K ) )  ->  ( ( r  e.  ( Atoms `  K
)  /\  s  e.  ( Atoms `  K )
)  ->  ( (
q  =/=  r  /\  -.  s  .<_  ( q ( join `  K
) r )  /\  Y  =  ( (
q ( join `  K
) r ) (
join `  K )
s ) )  ->  -.  X  .<_  Y ) ) )
2221rexlimdvv 2836 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  /\  q  e.  ( Atoms `  K ) )  ->  ( E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q ( join `  K
) r )  /\  Y  =  ( (
q ( join `  K
) r ) (
join `  K )
s ) )  ->  -.  X  .<_  Y ) )
2322rexlimdva 2830 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q (
join `  K )
r )  /\  Y  =  ( ( q ( join `  K
) r ) (
join `  K )
s ) )  ->  -.  X  .<_  Y ) )
2423adantld 454 . 2  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  ( ( Y  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) E. s  e.  ( Atoms `  K ) ( q  =/=  r  /\  -.  s  .<_  ( q (
join `  K )
r )  /\  Y  =  ( ( q ( join `  K
) r ) (
join `  K )
s ) ) )  ->  -.  X  .<_  Y ) )
259, 24mpd 15 1  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  -.  X  .<_  Y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Atomscatm 30061   HLchlt 30148   LPlanesclpl 30289   LVolsclvol 30290
This theorem is referenced by:  lvolnelpln  30387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297
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