Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lvolnlelpln Structured version   Unicode version

Theorem lvolnlelpln 35706
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 996 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  Y  e.  P )
2 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 lvolnlelpln.l . . . . 5  |-  .<_  =  ( le `  K )
4 eqid 2454 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2454 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 lvolnlelpln.p . . . . 5  |-  P  =  ( LPlanes `  K )
72, 3, 4, 5, 6islpln2 35657 . . . 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 1015 . . 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 1087 . . . . . . . 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 1088 . . . . . . . 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 1019 . . . . . . . 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 1020 . . . . . . . 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 1021 . . . . . . . 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 35703 . . . . . . . 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 1233 . . . . . . 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 1032 . . . . . . . 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 4451 . . . . . . 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 299 . . . . . 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 1193 . . . . 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 2952 . . . 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 2946 . . 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 465 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   lecple 14791   joincjn 15772   Atomscatm 35385   HLchlt 35472   LPlanesclpl 35613   LVolsclvol 35614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-lat 15875  df-clat 15937  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621
This theorem is referenced by:  lvolnelpln  35711
  Copyright terms: Public domain W3C validator