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Theorem lvolnlelpln 33069
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 1007 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  P )  ->  Y  e.  P )
2 eqid 2422 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 lvolnlelpln.l . . . . 5  |-  .<_  =  ( le `  K )
4 eqid 2422 . . . . 5  |-  ( join `  K )  =  (
join `  K )
5 eqid 2422 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
6 lvolnlelpln.p . . . . 5  |-  P  =  ( LPlanes `  K )
72, 3, 4, 5, 6islpln2 33020 . . . 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 1026 . . 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 213 . 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 1098 . . . . . . . 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 1099 . . . . . . . 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 1030 . . . . . . . 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 1031 . . . . . . . 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 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 ) ) )  ->  s  e.  ( Atoms `  K )
)
15 lvolnlelpln.v . . . . . . . . 9  |-  V  =  ( LVols `  K )
163, 4, 5, 15lvolnle3at 33066 . . . . . . . 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 1271 . . . . . . 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 1043 . . . . . . . 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 4432 . . . . . . 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 302 . . . . . 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 1204 . . . . 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 2923 . . . 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 2917 . . 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 468 . 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   E.wrex 2776   class class class wbr 4420   ` cfv 5598  (class class class)co 6302   Basecbs 15109   lecple 15185   joincjn 16177   Atomscatm 32748   HLchlt 32835   LPlanesclpl 32976   LVolsclvol 32977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-preset 16161  df-poset 16179  df-plt 16192  df-lub 16208  df-glb 16209  df-join 16210  df-meet 16211  df-p0 16273  df-lat 16280  df-clat 16342  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-llines 32982  df-lplanes 32983  df-lvols 32984
This theorem is referenced by:  lvolnelpln  33074
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