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Theorem islpln2a 33550
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 6210 . . . . . . . 8  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
2 islpln2a.j . . . . . . . . . 10  |-  .\/  =  ( join `  K )
3 islpln2a.a . . . . . . . . . 10  |-  A  =  ( Atoms `  K )
42, 3hlatjidm 33371 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
543ad2antr2 1154 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .\/  R )  =  R )
61, 5sylan9eqr 2517 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
76oveq1d 6218 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  (
( Q  .\/  R
)  .\/  S )  =  ( R  .\/  S ) )
8 simpll 753 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  K  e.  HL )
9 simplr2 1031 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  R  e.  A )
10 simplr3 1032 . . . . . . 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 33546 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  -.  ( R  .\/  S )  e.  P )
138, 9, 10, 12syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( R  .\/  S )  e.  P )
147, 13eqneltrd 2563 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P )
1514ex 434 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  =  R  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P ) )
1615necon2ad 2665 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  Q  =/=  R ) )
17 hllat 33366 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1817adantr 465 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
19 simpr3 996 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
20 eqid 2454 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2120, 3atbase 33292 . . . . . . 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 33369 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
24233adant3r3 1199 . . . . . 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 15356 . . . . . 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 1219 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
282, 3, 112atnelpln 33546 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  -.  ( Q  .\/  R )  e.  P )
29283adant3r3 1199 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( Q  .\/  R )  e.  P )
30 eleq1 2526 . . . . . . 7  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( (
( Q  .\/  R
)  .\/  S )  e.  P  <->  ( Q  .\/  R )  e.  P ) )
3130notbid 294 . . . . . 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 533 . 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 33539 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
37363expia 1190 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203   Basecbs 14295   lecple 14367   joincjn 15236   Latclat 15337   Atomscatm 33266   HLchlt 33353   LPlanesclpl 33494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-poset 15238  df-plt 15250  df-lub 15266  df-glb 15267  df-join 15268  df-meet 15269  df-p0 15331  df-lat 15338  df-clat 15400  df-oposet 33179  df-ol 33181  df-oml 33182  df-covers 33269  df-ats 33270  df-atl 33301  df-cvlat 33325  df-hlat 33354  df-llines 33500  df-lplanes 33501
This theorem is referenced by:  islpln2ah  33551  2atmat  33563  dalawlem13  33885  cdleme16d  34283
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