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Theorem islpln2a 35724
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 6225 . . . . . . . 8  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 islpln2a.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
3 islpln2a.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 35545 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
543ad2antr2 1160 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .\/  R )  =  R )
61, 5sylan9eqr 2459 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
76oveq1d 6233 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  (
( Q  .\/  R
)  .\/  S )  =  ( R  .\/  S ) )
8 simpll 751 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  K  e.  HL )
9 simplr2 1037 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  R  e.  A )
10 simplr3 1038 . . . . . . 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 35720 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  -.  ( R  .\/  S )  e.  P )
138, 9, 10, 12syl3anc 1226 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( R  .\/  S )  e.  P )
147, 13eqneltrd 2505 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P )
1514ex 432 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  =  R  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P ) )
1615necon2ad 2609 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  Q  =/=  R ) )
17 hllat 35540 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1817adantr 463 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
19 simpr3 1002 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
20 eqid 2396 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 3atbase 35466 . . . . . . 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 35543 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
24233adant3r3 1205 . . . . . 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 15834 . . . . . 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 1226 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
282, 3, 112atnelpln 35720 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  -.  ( Q  .\/  R )  e.  P )
29283adant3r3 1205 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( Q  .\/  R )  e.  P )
30 eleq1 2468 . . . . . . 7  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( (
( Q  .\/  R
)  .\/  S )  e.  P  <->  ( Q  .\/  R )  e.  P ) )
3130notbid 292 . . . . . 6  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( -.  ( ( Q  .\/  R )  .\/  S )  e.  P  <->  -.  ( Q  .\/  R )  e.  P ) )
3229, 31syl5ibrcom 222 . . . . 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 215 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  ->  -.  (
( Q  .\/  R
)  .\/  S )  e.  P ) )
3433con2d 115 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  -.  S  .<_  ( Q  .\/  R ) ) )
3516, 34jcad 531 . 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 35713 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
37363expia 1196 . 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 191 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2591   class class class wbr 4384   ` cfv 5513  (class class class)co 6218   Basecbs 14657   lecple 14732   joincjn 15713   Latclat 15815   Atomscatm 35440   HLchlt 35527   LPlanesclpl 35668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6180  df-ov 6221  df-oprab 6222  df-preset 15697  df-poset 15715  df-plt 15728  df-lub 15744  df-glb 15745  df-join 15746  df-meet 15747  df-p0 15809  df-lat 15816  df-clat 15878  df-oposet 35353  df-ol 35355  df-oml 35356  df-covers 35443  df-ats 35444  df-atl 35475  df-cvlat 35499  df-hlat 35528  df-llines 35674  df-lplanes 35675
This theorem is referenced by:  islpln2ah  35725  2atmat  35737  dalawlem13  36059  cdleme16d  36458
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