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Theorem lvolnlelpln 33538
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 990 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  Y  e.  P )
2 eqid 2451 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 lvolnlelpln.l . . . . 5  |-  .<_  =  ( le `  K )
4 eqid 2451 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2451 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 lvolnlelpln.p . . . . 5  |-  P  =  ( LPlanes `  K )
72, 3, 4, 5, 6islpln2 33489 . . . 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 1009 . . 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 210 . 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 1081 . . . . . . . 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 1082 . . . . . . . 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 1013 . . . . . . . 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 1014 . . . . . . . 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 1015 . . . . . . . 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 33535 . . . . . . . 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 1226 . . . . . . 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 1026 . . . . . . . 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 4405 . . . . . . 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 301 . . . . . 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 1187 . . . . 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 2946 . . . 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 2940 . . 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 467 . 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   E.wrex 2796   class class class wbr 4393   ` cfv 5519  (class class class)co 6193   Basecbs 14285   lecple 14356   joincjn 15225   Atomscatm 33217   HLchlt 33304   LPlanesclpl 33445   LVolsclvol 33446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-lat 15327  df-clat 15389  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453
This theorem is referenced by:  lvolnelpln  33543
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