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Theorem lplnnlelln 30354
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 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
3 eqid 2296 . . . . 5  |-  ( join `  K )  =  (
join `  K )
4 eqid 2296 . . . . 5  |-  ( Atoms `  K )  =  (
Atoms `  K )
5 lplnnlelln.n . . . . 5  |-  N  =  ( LLines `  K )
62, 3, 4, 5islln2 30322 . . . 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 30352 . . . . . . 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 4051 . . . . . 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 2686 . . 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 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Atomscatm 30075   HLchlt 30162   LLinesclln 30302   LPlanesclpl 30303
This theorem is referenced by:  lplnnelln  30357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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