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Theorem lplnnlelln 29732
Description: A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)
Hypotheses
Ref Expression
lplnnlelln.l  |-  .<_  =  ( le `  K )
lplnnlelln.n  |-  N  =  ( LLines `  K )
lplnnlelln.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplnnlelln  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  -.  X  .<_  Y )

Proof of Theorem lplnnlelln
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 957 . . 3  |-  ( ( K  e.  HL  /\  X  e.  P  /\  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 lplnnlelln.n . . . . 5  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln2 29700 . . . 4  |-  ( K  e.  HL  ->  ( Y  e.  N  <->  ( Y  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  Y  =  ( q
( join `  K )
r ) ) ) ) )
763ad2ant1 976 . . 3  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( Y  e.  N  <->  ( Y  e.  ( Base `  K )  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K )
( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) ) ) )
81, 7mpbid 201 . 2  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( Y  e.  (
Base `  K )  /\  E. q  e.  (
Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  Y  =  ( q (
join `  K )
r ) ) ) )
9 simp11 985 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  K  e.  HL )
10 simp12 986 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  X  e.  P )
11 simp2l 981 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  q  e.  ( Atoms `  K )
)
12 simp2r 982 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  r  e.  ( Atoms `  K )
)
13 lplnnlelln.l . . . . . . . 8  |-  .<_  =  ( le `  K )
14 lplnnlelln.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
1513, 3, 4, 14lplnnle2at 29730 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( X  e.  P  /\  q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K ) ) )  ->  -.  X  .<_  ( q ( join `  K
) r ) )
169, 10, 11, 12, 15syl13anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  -.  X  .<_  ( q ( join `  K ) r ) )
17 simp3r 984 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  Y  =  ( q ( join `  K ) r ) )
1817breq2d 4035 . . . . . 6  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  ( X  .<_  Y  <->  X  .<_  ( q ( join `  K
) r ) ) )
1916, 18mtbird 292 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  ( q  e.  (
Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  /\  ( q  =/=  r  /\  Y  =  (
q ( join `  K
) r ) ) )  ->  -.  X  .<_  Y )
20193exp 1150 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( ( q  e.  ( Atoms `  K )  /\  r  e.  ( Atoms `  K ) )  ->  ( ( q  =/=  r  /\  Y  =  ( q (
join `  K )
r ) )  ->  -.  X  .<_  Y ) ) )
2120rexlimdvv 2673 . . 3  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  Y  =  ( q
( join `  K )
r ) )  ->  -.  X  .<_  Y ) )
2221adantld 453 . 2  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( ( Y  e.  ( Base `  K
)  /\  E. q  e.  ( Atoms `  K ) E. r  e.  ( Atoms `  K ) ( q  =/=  r  /\  Y  =  ( q
( join `  K )
r ) ) )  ->  -.  X  .<_  Y ) )
238, 22mpd 14 1  |-  ( ( K  e.  HL  /\  X  e.  P  /\  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   LPlanesclpl 29681
This theorem is referenced by:  lplnnelln  29735
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
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