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Theorem islpln2a 29737
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 simpll 730 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  K  e.  HL )
2 simplr2 998 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  R  e.  A )
3 simplr3 999 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  S  e.  A )
4 islpln2a.j . . . . . . . 8  |-  .\/  =  ( join `  K )
5 islpln2a.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
6 islpln2a.p . . . . . . . 8  |-  P  =  ( LPlanes `  K )
74, 5, 62atnelpln 29733 . . . . . . 7  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  -.  ( R  .\/  S )  e.  P )
81, 2, 3, 7syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( R  .\/  S )  e.  P )
9 oveq1 5865 . . . . . . . . 9  |-  ( Q  =  R  ->  ( Q  .\/  R )  =  ( R  .\/  R
) )
104, 5hlatjidm 29558 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  R  e.  A )  ->  ( R  .\/  R
)  =  R )
11103ad2antr2 1121 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( R  .\/  R )  =  R )
129, 11sylan9eqr 2337 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  ( Q  .\/  R )  =  R )
1312oveq1d 5873 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  (
( Q  .\/  R
)  .\/  S )  =  ( R  .\/  S ) )
1413eleq1d 2349 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  <->  ( R  .\/  S )  e.  P
) )
158, 14mtbird 292 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  /\  Q  =  R )  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P )
1615ex 423 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  =  R  ->  -.  ( ( Q  .\/  R )  .\/  S )  e.  P ) )
1716necon2ad 2494 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  Q  =/=  R ) )
18 hllat 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1918adantr 451 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  K  e.  Lat )
20 simpr3 963 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  A )
21 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
2221, 5atbase 29479 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
2320, 22syl 15 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  S  e.  ( Base `  K
) )
2421, 4, 5hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
25243adant3r3 1162 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
26 islpln2a.l . . . . . . 7  |-  .<_  =  ( le `  K )
2721, 26, 4latleeqj2 14170 . . . . . 6  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
2819, 23, 25, 27syl3anc 1182 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  <->  ( ( Q 
.\/  R )  .\/  S )  =  ( Q 
.\/  R ) ) )
294, 5, 62atnelpln 29733 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  -.  ( Q  .\/  R )  e.  P )
30293adant3r3 1162 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  -.  ( Q  .\/  R )  e.  P )
31 eleq1 2343 . . . . . . 7  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( (
( Q  .\/  R
)  .\/  S )  e.  P  <->  ( Q  .\/  R )  e.  P ) )
3231notbid 285 . . . . . 6  |-  ( ( ( Q  .\/  R
)  .\/  S )  =  ( Q  .\/  R )  ->  ( -.  ( ( Q  .\/  R )  .\/  S )  e.  P  <->  -.  ( Q  .\/  R )  e.  P ) )
3330, 32syl5ibrcom 213 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( Q  .\/  R )  ->  -.  (
( Q  .\/  R
)  .\/  S )  e.  P ) )
3428, 33sylbid 206 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  ( S  .<_  ( Q  .\/  R )  ->  -.  (
( Q  .\/  R
)  .\/  S )  e.  P ) )
3534con2d 107 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  -.  S  .<_  ( Q  .\/  R ) ) )
3617, 35jcad 519 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  ->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R ) ) ) )
3726, 4, 5, 6lplni2 29726 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
38373expia 1153 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )  ->  (
( Q  =/=  R  /\  -.  S  .<_  ( Q 
.\/  R ) )  ->  ( ( Q 
.\/  R )  .\/  S )  e.  P ) )
3936, 38impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   Latclat 14151   Atomscatm 29453   HLchlt 29540   LPlanesclpl 29681
This theorem is referenced by:  islpln2ah  29738  2atmat  29750  dalawlem13  30072  cdleme16d  30470
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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