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Theorem islpln2a 30346
Description: The predicate "is a lattice plane" for join of atoms. (Contributed by NM, 16-Jul-2012.)
Hypotheses
Ref Expression
islpln2a.l  |-  .<_  =  ( le `  K )
islpln2a.j  |-  .\/  =  ( join `  K )
islpln2a.a  |-  A  =  ( Atoms `  K )
islpln2a.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
islpln2a  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  <->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) ) )

Proof of Theorem islpln2a
StepHypRef Expression
1 oveq1 6089 . . . . . . . 8  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 islpln2a.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
3 islpln2a.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 30167 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
543ad2antr2 1124 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .\/  R )  =  R )
61, 5sylan9eqr 2491 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
76oveq1d 6097 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  (
( Q  .\/  R
)  .\/  S )  =  ( R  .\/  S ) )
8 simpll 732 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  K  e.  HL )
9 simplr2 1001 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  R  e.  A )
10 simplr3 1002 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  S  e.  A )
11 islpln2a.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
122, 3, 112atnelpln 30342 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  -.  ( R  .\/  S )  e.  P )
138, 9, 10, 12syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( R  .\/  S )  e.  P )
147, 13eqneltrd 2530 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P )
1514ex 425 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  =  R  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P ) )
1615necon2ad 2653 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  Q  =/=  R ) )
17 hllat 30162 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1817adantr 453 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
19 simpr3 966 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
20 eqid 2437 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 3atbase 30088 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
2219, 21syl 16 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
2320, 2, 3hlatjcl 30165 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
24233adant3r3 1165 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
25 islpln2a.l . . . . . . 7  |-  .<_  =  ( le `  K )
2620, 25, 2latleeqj2 14494 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
2718, 22, 24, 26syl3anc 1185 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
282, 3, 112atnelpln 30342 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  -.  ( Q  .\/  R )  e.  P )
29283adant3r3 1165 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( Q  .\/  R )  e.  P )
30 eleq1 2497 . . . . . . 7  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( (
( Q  .\/  R
)  .\/  S )  e.  P  <->  ( Q  .\/  R )  e.  P ) )
3130notbid 287 . . . . . 6  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( -.  ( ( Q  .\/  R )  .\/  S )  e.  P  <->  -.  ( Q  .\/  R )  e.  P ) )
3229, 31syl5ibrcom 215 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( Q  .\/  R )  ->  -.  (
( Q  .\/  R
)  .\/  S )  e.  P ) )
3327, 32sylbid 208 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  ->  -.  (
( Q  .\/  R
)  .\/  S )  e.  P ) )
3433con2d 110 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  -.  S  .<_  ( Q  .\/  R ) ) )
3516, 34jcad 521 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R ) ) ) )
3625, 2, 3, 11lplni2 30335 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
37363expia 1156 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  S )  e.  P ) )
3835, 37impbid 185 1  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  <->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   Latclat 14475   Atomscatm 30062   HLchlt 30149   LPlanesclpl 30290
This theorem is referenced by:  islpln2ah  30347  2atmat  30359  dalawlem13  30681  cdleme16d  31079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-llines 30296  df-lplanes 30297
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