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Theorem lplni2 32534
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 998 . . 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 1025 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  =/=  R )
3 simp3r 1026 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  S  .<_  ( Q 
.\/  R ) )
4 eqidd 2403 . . 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 2684 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
6 oveq1 6284 . . . . . . 7  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
76breq2d 4406 . . . . . 6  |-  ( q  =  Q  ->  (
s  .<_  ( q  .\/  r )  <->  s  .<_  ( Q  .\/  r ) ) )
87notbid 292 . . . . 5  |-  ( q  =  Q  ->  ( -.  s  .<_  ( q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  r
) ) )
96oveq1d 6292 . . . . . 6  |-  ( q  =  Q  ->  (
( q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  r )  .\/  s ) )
109eqeq2d 2416 . . . . 5  |-  ( q  =  Q  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) )
115, 8, 103anbi123d 1301 . . . 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 2686 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
13 oveq2 6285 . . . . . . 7  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1413breq2d 4406 . . . . . 6  |-  ( r  =  R  ->  (
s  .<_  ( Q  .\/  r )  <->  s  .<_  ( Q  .\/  R ) ) )
1514notbid 292 . . . . 5  |-  ( r  =  R  ->  ( -.  s  .<_  ( Q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  R
) ) )
1613oveq1d 6292 . . . . . 6  |-  ( r  =  R  ->  (
( Q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  s ) )
1716eqeq2d 2416 . . . . 5  |-  ( r  =  R  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) )
1812, 15, 173anbi123d 1301 . . . 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 4397 . . . . . 6  |-  ( s  =  S  ->  (
s  .<_  ( Q  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
2019notbid 292 . . . . 5  |-  ( s  =  S  ->  ( -.  s  .<_  ( Q 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
21 oveq2 6285 . . . . . 6  |-  ( s  =  S  ->  (
( Q  .\/  R
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  S ) )
2221eqeq2d 2416 . . . . 5  |-  ( s  =  S  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2320, 223anbi23d 1304 . . . 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 3172 . . 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 1232 . 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 997 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  HL )
27 hllat 32361 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
28273ad2ant1 1018 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  Lat )
29 simp21 1030 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  e.  A )
30 simp22 1031 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  R  e.  A )
31 eqid 2402 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
32 lplni2.j . . . . . 6  |-  .\/  =  ( join `  K )
33 lplni2.a . . . . . 6  |-  A  =  ( Atoms `  K )
3431, 32, 33hlatjcl 32364 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3526, 29, 30, 34syl3anc 1230 . . . 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 1032 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  A )
3731, 33atbase 32287 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
3836, 37syl 17 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  ( Base `  K ) )
3931, 32latjcl 16003 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
4028, 35, 38, 39syl3anc 1230 . . 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 32532 . . 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 659 . 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 232 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2754   class class class wbr 4394   ` cfv 5568  (class class class)co 6277   Basecbs 14839   lecple 14914   joincjn 15895   Latclat 15997   Atomscatm 32261   HLchlt 32348   LPlanesclpl 32489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496
This theorem is referenced by:  islpln2a  32545  2llnjaN  32563  lvolnle3at  32579  dalem42  32711  cdleme16aN  33257
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