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Theorem lplnexatN 29752
Description: Given a lattice line on a lattice plane, there is an atom whose join with the line equals the plane. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
lplnexat.l  |-  .<_  =  ( le `  K )
lplnexat.j  |-  .\/  =  ( join `  K )
lplnexat.a  |-  A  =  ( Atoms `  K )
lplnexat.n  |-  N  =  ( LLines `  K )
lplnexat.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplnexatN  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  E. q  e.  A  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) )
Distinct variable groups:    A, q    K, q    .<_ , q    Y, q    X, q
Allowed substitution hints:    P( q)    .\/ ( q)    N( q)

Proof of Theorem lplnexatN
StepHypRef Expression
1 simp1 955 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  K  e.  HL )
2 simp3 957 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  Y  e.  N )
3 simp2 956 . . . 4  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  X  e.  P )
41, 2, 33jca 1132 . . 3  |-  ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  ->  ( K  e.  HL  /\  Y  e.  N  /\  X  e.  P )
)
5 lplnexat.l . . . 4  |-  .<_  =  ( le `  K )
6 eqid 2283 . . . 4  |-  (  <o  `  K )  =  ( 
<o  `  K )
7 lplnexat.n . . . 4  |-  N  =  ( LLines `  K )
8 lplnexat.p . . . 4  |-  P  =  ( LPlanes `  K )
95, 6, 7, 8llncvrlpln2 29746 . . 3  |-  ( ( ( K  e.  HL  /\  Y  e.  N  /\  X  e.  P )  /\  Y  .<_  X )  ->  Y (  <o  `  K ) X )
104, 9sylan 457 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  Y (  <o  `  K ) X )
11 simpl1 958 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  K  e.  HL )
12 simpl3 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  Y  e.  N
)
13 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
1413, 7llnbase 29698 . . . . 5  |-  ( Y  e.  N  ->  Y  e.  ( Base `  K
) )
1512, 14syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  Y  e.  (
Base `  K )
)
16 simpl2 959 . . . . 5  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  X  e.  P
)
1713, 8lplnbase 29723 . . . . 5  |-  ( X  e.  P  ->  X  e.  ( Base `  K
) )
1816, 17syl 15 . . . 4  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  X  e.  (
Base `  K )
)
19 lplnexat.j . . . . 5  |-  .\/  =  ( join `  K )
20 lplnexat.a . . . . 5  |-  A  =  ( Atoms `  K )
2113, 5, 19, 6, 20cvrval3 29602 . . . 4  |-  ( ( K  e.  HL  /\  Y  e.  ( Base `  K )  /\  X  e.  ( Base `  K
) )  ->  ( Y (  <o  `  K
) X  <->  E. q  e.  A  ( -.  q  .<_  Y  /\  ( Y  .\/  q )  =  X ) ) )
2211, 15, 18, 21syl3anc 1182 . . 3  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  ( Y ( 
<o  `  K ) X  <->  E. q  e.  A  ( -.  q  .<_  Y  /\  ( Y  .\/  q )  =  X ) ) )
23 eqcom 2285 . . . . 5  |-  ( ( Y  .\/  q )  =  X  <->  X  =  ( Y  .\/  q ) )
2423anbi2i 675 . . . 4  |-  ( ( -.  q  .<_  Y  /\  ( Y  .\/  q )  =  X )  <->  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) )
2524rexbii 2568 . . 3  |-  ( E. q  e.  A  ( -.  q  .<_  Y  /\  ( Y  .\/  q )  =  X )  <->  E. q  e.  A  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) )
2622, 25syl6bb 252 . 2  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  ( Y ( 
<o  `  K ) X  <->  E. q  e.  A  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) ) )
2710, 26mpbid 201 1  |-  ( ( ( K  e.  HL  /\  X  e.  P  /\  Y  e.  N )  /\  Y  .<_  X )  ->  E. q  e.  A  ( -.  q  .<_  Y  /\  X  =  ( Y  .\/  q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078    <o ccvr 29452   Atomscatm 29453   HLchlt 29540   LLinesclln 29680   LPlanesclpl 29681
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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