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Theorem lplni2 30334
Description: The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012.)
Hypotheses
Ref Expression
lplni2.l  |-  .<_  =  ( le `  K )
lplni2.j  |-  .\/  =  ( join `  K )
lplni2.a  |-  A  =  ( Atoms `  K )
lplni2.p  |-  P  =  ( LPlanes `  K )
Assertion
Ref Expression
lplni2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )

Proof of Theorem lplni2
Dummy variables  r 
q  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 958 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( Q  e.  A  /\  R  e.  A  /\  S  e.  A
) )
2 simp3l 985 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  =/=  R )
3 simp3r 986 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  S  .<_  ( Q 
.\/  R ) )
4 eqidd 2437 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  S ) )
5 neeq1 2609 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
6 oveq1 6088 . . . . . . 7  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
76breq2d 4224 . . . . . 6  |-  ( q  =  Q  ->  (
s  .<_  ( q  .\/  r )  <->  s  .<_  ( Q  .\/  r ) ) )
87notbid 286 . . . . 5  |-  ( q  =  Q  ->  ( -.  s  .<_  ( q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  r
) ) )
96oveq1d 6096 . . . . . 6  |-  ( q  =  Q  ->  (
( q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  r )  .\/  s ) )
109eqeq2d 2447 . . . . 5  |-  ( q  =  Q  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) )
115, 8, 103anbi123d 1254 . . . 4  |-  ( q  =  Q  ->  (
( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s ) )  <->  ( Q  =/=  r  /\  -.  s  .<_  ( Q  .\/  r
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) ) )
12 neeq2 2610 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
13 oveq2 6089 . . . . . . 7  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1413breq2d 4224 . . . . . 6  |-  ( r  =  R  ->  (
s  .<_  ( Q  .\/  r )  <->  s  .<_  ( Q  .\/  R ) ) )
1514notbid 286 . . . . 5  |-  ( r  =  R  ->  ( -.  s  .<_  ( Q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  R
) ) )
1613oveq1d 6096 . . . . . 6  |-  ( r  =  R  ->  (
( Q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  s ) )
1716eqeq2d 2447 . . . . 5  |-  ( r  =  R  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) )
1812, 15, 173anbi123d 1254 . . . 4  |-  ( r  =  R  ->  (
( Q  =/=  r  /\  -.  s  .<_  ( Q 
.\/  r )  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s ) )  <->  ( Q  =/=  R  /\  -.  s  .<_  ( Q  .\/  R
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) ) )
19 breq1 4215 . . . . . 6  |-  ( s  =  S  ->  (
s  .<_  ( Q  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
2019notbid 286 . . . . 5  |-  ( s  =  S  ->  ( -.  s  .<_  ( Q 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
21 oveq2 6089 . . . . . 6  |-  ( s  =  S  ->  (
( Q  .\/  R
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  S ) )
2221eqeq2d 2447 . . . . 5  |-  ( s  =  S  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2320, 223anbi23d 1257 . . . 4  |-  ( s  =  S  ->  (
( Q  =/=  R  /\  -.  s  .<_  ( Q 
.\/  R )  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s ) )  <->  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) ) )
2411, 18, 23rspc3ev 3062 . . 3  |-  ( ( ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( q  .\/  r ) 
.\/  s ) ) )
251, 2, 3, 4, 24syl13anc 1186 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( q  =/=  r  /\  -.  s  .<_  ( q 
.\/  r )  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s ) ) )
26 simp1 957 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  HL )
27 hllat 30161 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
28273ad2ant1 978 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  Lat )
29 simp21 990 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  e.  A )
30 simp22 991 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  R  e.  A )
31 eqid 2436 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
32 lplni2.j . . . . . 6  |-  .\/  =  ( join `  K )
33 lplni2.a . . . . . 6  |-  A  =  ( Atoms `  K )
3431, 32, 33hlatjcl 30164 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3526, 29, 30, 34syl3anc 1184 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( Q  .\/  R
)  e.  ( Base `  K ) )
36 simp23 992 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  A )
3731, 33atbase 30087 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
3836, 37syl 16 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  ( Base `  K ) )
3931, 32latjcl 14479 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
4028, 35, 38, 39syl3anc 1184 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K
) )
41 lplni2.l . . . 4  |-  .<_  =  ( le `  K )
42 lplni2.p . . . 4  |-  P  =  ( LPlanes `  K )
4331, 41, 32, 33, 42islpln5 30332 . . 3  |-  ( ( K  e.  HL  /\  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K
) )  ->  (
( ( Q  .\/  R )  .\/  S )  e.  P  <->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( q  .\/  r ) 
.\/  s ) ) ) )
4426, 40, 43syl2anc 643 . 2  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( ( Q 
.\/  R )  .\/  S )  e.  P  <->  E. q  e.  A  E. r  e.  A  E. s  e.  A  ( q  =/=  r  /\  -.  s  .<_  ( q  .\/  r
)  /\  ( ( Q  .\/  R )  .\/  S )  =  ( ( q  .\/  r ) 
.\/  s ) ) ) )
4525, 44mpbird 224 1  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  -> 
( ( Q  .\/  R )  .\/  S )  e.  P )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   Latclat 14474   Atomscatm 30061   HLchlt 30148   LPlanesclpl 30289
This theorem is referenced by:  islpln2a  30345  2llnjaN  30363  lvolnle3at  30379  dalem42  30511  cdleme16aN  31056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296
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