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Theorem lvolnlelln 29591
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 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  V  /\  Y  e.  N )  ->  Y  e.  N )
2 eqid 2316 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2316 . . . . 5  |-  ( join `  K )  =  (
join `  K )
4 eqid 2316 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 lvolnlelln.n . . . . 5  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln2 29518 . . . 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 976 . . 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 201 . 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 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 ) ) )  ->  K  e.  HL )
10 simp12 986 . . . . . . 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 981 . . . . . . 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 982 . . . . . . 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 29589 . . . . . . 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 1189 . . . . . 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 984 . . . . . . . 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 29376 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  p  e.  ( Atoms `  K ) )  -> 
( p ( join `  K ) p )  =  p )
199, 11, 18syl2anc 642 . . . . . . . . 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 5915 . . . . . . . 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 2351 . . . . . . 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 4072 . . . . . 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 292 . . . . 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 1150 . . . 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 2707 . . 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 453 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578   class class class wbr 4060   ` cfv 5292  (class class class)co 5900   Basecbs 13195   lecple 13262   joincjn 14127   Atomscatm 29271   HLchlt 29358   LLinesclln 29498   LVolsclvol 29500
This theorem is referenced by:  lvolnelln  29596
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-undef 6340  df-riota 6346  df-poset 14129  df-plt 14141  df-lub 14157  df-glb 14158  df-join 14159  df-meet 14160  df-p0 14194  df-lat 14201  df-clat 14263  df-oposet 29184  df-ol 29186  df-oml 29187  df-covers 29274  df-ats 29275  df-atl 29306  df-cvlat 29330  df-hlat 29359  df-llines 29505  df-lplanes 29506  df-lvols 29507
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