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Theorem lplni2 33179
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 989 . . 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 1016 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  =/=  R )
3 simp3r 1017 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  S  .<_  ( Q 
.\/  R ) )
4 eqidd 2443 . . 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 2615 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
6 oveq1 6097 . . . . . . 7  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
76breq2d 4303 . . . . . 6  |-  ( q  =  Q  ->  (
s  .<_  ( q  .\/  r )  <->  s  .<_  ( Q  .\/  r ) ) )
87notbid 294 . . . . 5  |-  ( q  =  Q  ->  ( -.  s  .<_  ( q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  r
) ) )
96oveq1d 6105 . . . . . 6  |-  ( q  =  Q  ->  (
( q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  r )  .\/  s ) )
109eqeq2d 2453 . . . . 5  |-  ( q  =  Q  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) )
115, 8, 103anbi123d 1289 . . . 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 2616 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
13 oveq2 6098 . . . . . . 7  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1413breq2d 4303 . . . . . 6  |-  ( r  =  R  ->  (
s  .<_  ( Q  .\/  r )  <->  s  .<_  ( Q  .\/  R ) ) )
1514notbid 294 . . . . 5  |-  ( r  =  R  ->  ( -.  s  .<_  ( Q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  R
) ) )
1613oveq1d 6105 . . . . . 6  |-  ( r  =  R  ->  (
( Q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  s ) )
1716eqeq2d 2453 . . . . 5  |-  ( r  =  R  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) )
1812, 15, 173anbi123d 1289 . . . 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 4294 . . . . . 6  |-  ( s  =  S  ->  (
s  .<_  ( Q  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
2019notbid 294 . . . . 5  |-  ( s  =  S  ->  ( -.  s  .<_  ( Q 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
21 oveq2 6098 . . . . . 6  |-  ( s  =  S  ->  (
( Q  .\/  R
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  S ) )
2221eqeq2d 2453 . . . . 5  |-  ( s  =  S  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2320, 223anbi23d 1292 . . . 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 3082 . . 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 1220 . 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 988 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  HL )
27 hllat 33006 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
28273ad2ant1 1009 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  Lat )
29 simp21 1021 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  e.  A )
30 simp22 1022 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  R  e.  A )
31 eqid 2442 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
32 lplni2.j . . . . . 6  |-  .\/  =  ( join `  K )
33 lplni2.a . . . . . 6  |-  A  =  ( Atoms `  K )
3431, 32, 33hlatjcl 33009 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3526, 29, 30, 34syl3anc 1218 . . . 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 1023 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  A )
3731, 33atbase 32932 . . . . 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 15220 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
4028, 35, 38, 39syl3anc 1218 . . 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 33177 . . 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 661 . 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2605   E.wrex 2715   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   lecple 14244   joincjn 15113   Latclat 15214   Atomscatm 32906   HLchlt 32993   LPlanesclpl 33134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-llines 33140  df-lplanes 33141
This theorem is referenced by:  islpln2a  33190  2llnjaN  33208  lvolnle3at  33224  dalem42  33356  cdleme16aN  33901
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