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Theorem lplni2 30348
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 956 . . 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 983 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  =/=  R )
3 simp3r 984 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  -.  S  .<_  ( Q 
.\/  R ) )
4 eqidd 2297 . . 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 2467 . . . . 5  |-  ( q  =  Q  ->  (
q  =/=  r  <->  Q  =/=  r ) )
6 oveq1 5881 . . . . . . 7  |-  ( q  =  Q  ->  (
q  .\/  r )  =  ( Q  .\/  r ) )
76breq2d 4051 . . . . . 6  |-  ( q  =  Q  ->  (
s  .<_  ( q  .\/  r )  <->  s  .<_  ( Q  .\/  r ) ) )
87notbid 285 . . . . 5  |-  ( q  =  Q  ->  ( -.  s  .<_  ( q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  r
) ) )
96oveq1d 5889 . . . . . 6  |-  ( q  =  Q  ->  (
( q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  r )  .\/  s ) )
109eqeq2d 2307 . . . . 5  |-  ( q  =  Q  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  r ) 
.\/  s ) ) )
115, 8, 103anbi123d 1252 . . . 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 2468 . . . . 5  |-  ( r  =  R  ->  ( Q  =/=  r  <->  Q  =/=  R ) )
13 oveq2 5882 . . . . . . 7  |-  ( r  =  R  ->  ( Q  .\/  r )  =  ( Q  .\/  R
) )
1413breq2d 4051 . . . . . 6  |-  ( r  =  R  ->  (
s  .<_  ( Q  .\/  r )  <->  s  .<_  ( Q  .\/  R ) ) )
1514notbid 285 . . . . 5  |-  ( r  =  R  ->  ( -.  s  .<_  ( Q 
.\/  r )  <->  -.  s  .<_  ( Q  .\/  R
) ) )
1613oveq1d 5889 . . . . . 6  |-  ( r  =  R  ->  (
( Q  .\/  r
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  s ) )
1716eqeq2d 2307 . . . . 5  |-  ( r  =  R  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  r )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  s ) ) )
1812, 15, 173anbi123d 1252 . . . 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 4042 . . . . . 6  |-  ( s  =  S  ->  (
s  .<_  ( Q  .\/  R )  <->  S  .<_  ( Q 
.\/  R ) ) )
2019notbid 285 . . . . 5  |-  ( s  =  S  ->  ( -.  s  .<_  ( Q 
.\/  R )  <->  -.  S  .<_  ( Q  .\/  R
) ) )
21 oveq2 5882 . . . . . 6  |-  ( s  =  S  ->  (
( Q  .\/  R
)  .\/  s )  =  ( ( Q 
.\/  R )  .\/  S ) )
2221eqeq2d 2307 . . . . 5  |-  ( s  =  S  ->  (
( ( Q  .\/  R )  .\/  S )  =  ( ( Q 
.\/  R )  .\/  s )  <->  ( ( Q  .\/  R )  .\/  S )  =  ( ( Q  .\/  R ) 
.\/  S ) ) )
2320, 223anbi23d 1255 . . . 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 2907 . . 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 1184 . 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 955 . . 3  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  HL )
27 hllat 30175 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
28273ad2ant1 976 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  K  e.  Lat )
29 simp21 988 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  Q  e.  A )
30 simp22 989 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  R  e.  A )
31 eqid 2296 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
32 lplni2.j . . . . . 6  |-  .\/  =  ( join `  K )
33 lplni2.a . . . . . 6  |-  A  =  ( Atoms `  K )
3431, 32, 33hlatjcl 30178 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
3526, 29, 30, 34syl3anc 1182 . . . 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 990 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  A )
3731, 33atbase 30101 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
3836, 37syl 15 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  S  e.  A
)  /\  ( Q  =/=  R  /\  -.  S  .<_  ( Q  .\/  R
) ) )  ->  S  e.  ( Base `  K ) )
3931, 32latjcl 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  .\/  S )  e.  ( Base `  K ) )
4028, 35, 38, 39syl3anc 1182 . . 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 30346 . . 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 642 . 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 223 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162   LPlanesclpl 30303
This theorem is referenced by:  islpln2a  30359  2llnjaN  30377  lvolnle3at  30393  dalem42  30525  cdleme16aN  31070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310
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