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Theorem lplni2 34210
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 992 . . 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 1019 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  =/=  R )
3 simp3r 1020 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  S  .<_  ( Q 
.\/  R ) )
4 eqidd 2463 . . 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 2743 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
6 oveq1 6284 . . . . . . 7  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
76breq2d 4454 . . . . . 6  |-  ( q  =  Q  ->  (
s  .<_  ( q  .\/  r )  <->  s  .<_  ( Q  .\/  r ) ) )
87notbid 294 . . . . 5  |-  ( q  =  Q  ->  ( -.  s  .<_  ( q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  r
) ) )
96oveq1d 6292 . . . . . 6  |-  ( q  =  Q  ->  (
( q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  r )  .\/  s ) )
109eqeq2d 2476 . . . . 5  |-  ( q  =  Q  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) )
115, 8, 103anbi123d 1294 . . . 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 2745 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
13 oveq2 6285 . . . . . . 7  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1413breq2d 4454 . . . . . 6  |-  ( r  =  R  ->  (
s  .<_  ( Q  .\/  r )  <->  s  .<_  ( Q  .\/  R ) ) )
1514notbid 294 . . . . 5  |-  ( r  =  R  ->  ( -.  s  .<_  ( Q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  R
) ) )
1613oveq1d 6292 . . . . . 6  |-  ( r  =  R  ->  (
( Q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  s ) )
1716eqeq2d 2476 . . . . 5  |-  ( r  =  R  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) )
1812, 15, 173anbi123d 1294 . . . 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 4445 . . . . . 6  |-  ( s  =  S  ->  (
s  .<_  ( Q  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
2019notbid 294 . . . . 5  |-  ( s  =  S  ->  ( -.  s  .<_  ( Q 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
21 oveq2 6285 . . . . . 6  |-  ( s  =  S  ->  (
( Q  .\/  R
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  S ) )
2221eqeq2d 2476 . . . . 5  |-  ( s  =  S  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2320, 223anbi23d 1297 . . . 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 3222 . . 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 1225 . 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 991 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  HL )
27 hllat 34037 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
28273ad2ant1 1012 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  Lat )
29 simp21 1024 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  e.  A )
30 simp22 1025 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  R  e.  A )
31 eqid 2462 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
32 lplni2.j . . . . . 6  |-  .\/  =  ( join `  K )
33 lplni2.a . . . . . 6  |-  A  =  ( Atoms `  K )
3431, 32, 33hlatjcl 34040 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3526, 29, 30, 34syl3anc 1223 . . . 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 1026 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  A )
3731, 33atbase 33963 . . . . 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 15529 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
4028, 35, 38, 39syl3anc 1223 . . 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 34208 . . 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 968    = wceq 1374    e. wcel 1762    =/= wne 2657   E.wrex 2810   class class class wbr 4442   ` cfv 5581  (class class class)co 6277   Basecbs 14481   lecple 14553   joincjn 15422   Latclat 15523   Atomscatm 33937   HLchlt 34024   LPlanesclpl 34165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-llines 34171  df-lplanes 34172
This theorem is referenced by:  islpln2a  34221  2llnjaN  34239  lvolnle3at  34255  dalem42  34387  cdleme16aN  34932
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