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Theorem lvolnlelln 29773
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 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2283 . . . . 5  |-  ( join `  K )  =  (
join `  K )
4 eqid 2283 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 lvolnlelln.n . . . . 5  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln2 29700 . . . 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 29771 . . . . . . 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 29558 . . . . . . . . . 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 5873 . . . . . . . 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 2318 . . . . . . 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 4035 . . . . . 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 2673 . . 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 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LVolsclvol 29682
This theorem is referenced by:  lvolnelln  29778
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689
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