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Theorem 2atm 30251
Description: An atom majorized by two different atom joins (which could be atoms or lines) is equal to their intersection. (Contributed by NM, 30-Jun-2013.)
Hypotheses
Ref Expression
2atm.l  |-  .<_  =  ( le `  K )
2atm.j  |-  .\/  =  ( join `  K )
2atm.m  |-  ./\  =  ( meet `  K )
2atm.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2atm  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )

Proof of Theorem 2atm
StepHypRef Expression
1 simp31 993 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( P  .\/  Q ) )
2 simp32 994 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( R  .\/  S ) )
3 simp11 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  HL )
4 hllat 30088 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  Lat )
6 simp23 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  A )
7 eqid 2435 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 2atm.a . . . . . 6  |-  A  =  ( Atoms `  K )
97, 8atbase 30014 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
106, 9syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  e.  ( Base `  K )
)
11 simp12 988 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  A )
127, 8atbase 30014 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1311, 12syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  P  e.  ( Base `  K )
)
14 simp13 989 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  A )
157, 8atbase 30014 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
1614, 15syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  ( Base `  K )
)
17 2atm.j . . . . . 6  |-  .\/  =  ( join `  K )
187, 17latjcl 14471 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
195, 13, 16, 18syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
20 simp21 990 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
21 simp22 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  S  e.  A )
227, 17, 8hlatjcl 30091 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
233, 20, 21, 22syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
24 2atm.l . . . . 5  |-  .<_  =  ( le `  K )
25 2atm.m . . . . 5  |-  ./\  =  ( meet `  K )
267, 24, 25latlem12 14499 . . . 4  |-  ( ( K  e.  Lat  /\  ( T  e.  ( Base `  K )  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
) )  ->  (
( T  .<_  ( P 
.\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
275, 10, 19, 23, 26syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( T  .<_  ( P  .\/  Q )  /\  T  .<_  ( R  .\/  S ) )  <->  T  .<_  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) ) ) )
281, 2, 27mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  .<_  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) ) )
29 hlatl 30085 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
303, 29syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  AtLat
)
317, 25latmcl 14472 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( R  .\/  S )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
325, 19, 23, 31syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K ) )
33 eqid 2435 . . . . . . 7  |-  ( 0.
`  K )  =  ( 0. `  K
)
347, 24, 33, 8atlen0 30035 . . . . . 6  |-  ( ( ( K  e.  AtLat  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  ( Base `  K
)  /\  T  e.  A )  /\  T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )  ->  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3530, 32, 6, 28, 34syl31anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/=  ( 0.
`  K ) )
3635neneqd 2614 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  -.  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =  ( 0. `  K
) )
37 simp33 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  =/=  ( R  .\/  S ) )
3817, 25, 33, 82atmat0 30250 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
393, 11, 14, 20, 21, 37, 38syl33anc 1199 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  ( 0. `  K ) ) )
4039ord 367 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( -.  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( 0.
`  K ) ) )
4136, 40mt3d 119 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
4224, 8atcmp 30036 . . 3  |-  ( ( K  e.  AtLat  /\  T  e.  A  /\  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A )  ->  ( T  .<_  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) ) ) )
4330, 6, 41, 42syl3anc 1184 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( T  .<_  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  <->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) ) )
4428, 43mpbid 202 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  T  e.  A
)  /\  ( T  .<_  ( P  .\/  Q
)  /\  T  .<_  ( R  .\/  S )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  T  =  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13461   lecple 13528   joincjn 14393   meetcmee 14394   0.cp0 14458   Latclat 14466   Atomscatm 29988   AtLatcal 29989   HLchlt 30075
This theorem is referenced by:  cdlemk12  31574  cdlemk12u  31596  cdlemk47  31673
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-undef 6535  df-riota 6541  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-lat 14467  df-clat 14529  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222
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