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Theorem 2at0mat0 32522
Description: Special case of 2atmat0 32523 where one atom could be zero. (Contributed by NM, 30-May-2013.)
Hypotheses
Ref Expression
2atmatz.j  |-  .\/  =  ( join `  K )
2atmatz.m  |-  ./\  =  ( meet `  K )
2atmatz.z  |-  .0.  =  ( 0. `  K )
2atmatz.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
2at0mat0  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )

Proof of Theorem 2at0mat0
StepHypRef Expression
1 simpll 752 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  e.  A )  ->  ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A ) )
2 simplr1 1039 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  e.  A )  ->  R  e.  A )
3 simpr 459 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  e.  A )  ->  S  e.  A )
4 simplr3 1041 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  e.  A )  ->  ( P  .\/  Q )  =/=  ( R  .\/  S
) )
5 simpl1 1000 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  K  e.  HL )
6 hlol 32359 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
75, 6syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  K  e.  OL )
8 simpr1 1003 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
9 simpr2 1004 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  S  e.  A )
10 eqid 2402 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
11 2atmatz.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
12 2atmatz.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1310, 11, 12hlatjcl 32364 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
145, 8, 9, 13syl3anc 1230 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( R  .\/  S )  e.  (
Base `  K )
)
15 simpl3 1002 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  A )
16 2atmatz.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
17 2atmatz.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
1810, 16, 17, 12meetat2 32295 . . . . . . 7  |-  ( ( K  e.  OL  /\  ( R  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  A )  ->  (
( ( R  .\/  S )  ./\  Q )  e.  A  \/  (
( R  .\/  S
)  ./\  Q )  =  .0.  ) )
197, 14, 15, 18syl3anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( R  .\/  S
)  ./\  Q )  e.  A  \/  (
( R  .\/  S
)  ./\  Q )  =  .0.  ) )
2019adantr 463 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( ( R 
.\/  S )  ./\  Q )  e.  A  \/  ( ( R  .\/  S )  ./\  Q )  =  .0.  ) )
21 oveq1 6284 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2211, 12hlatjidm 32366 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
235, 15, 22syl2anc 659 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( Q  .\/  Q )  =  Q )
2421, 23sylan9eqr 2465 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( P  .\/  Q
)  =  Q )
2524oveq1d 6292 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( Q  ./\  ( R  .\/  S ) ) )
26 hllat 32361 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
275, 26syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  K  e.  Lat )
2810, 12atbase 32287 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2915, 28syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  Q  e.  ( Base `  K )
)
3010, 16latmcom 16027 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  S
)  ./\  Q )
)
3127, 29, 14, 30syl3anc 1230 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( Q  ./\  ( R  .\/  S
) )  =  ( ( R  .\/  S
)  ./\  Q )
)
3231adantr 463 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( Q  ./\  ( R  .\/  S ) )  =  ( ( R 
.\/  S )  ./\  Q ) )
3325, 32eqtrd 2443 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  S )  ./\  Q )
)
3433eleq1d 2471 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  <->  ( ( R  .\/  S )  ./\  Q )  e.  A ) )
3533eqeq1d 2404 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  <->  ( ( R  .\/  S )  ./\  Q )  =  .0.  )
)
3634, 35orbi12d 708 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )  <->  ( (
( R  .\/  S
)  ./\  Q )  e.  A  \/  (
( R  .\/  S
)  ./\  Q )  =  .0.  ) ) )
3720, 36mpbird 232 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =  Q )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
3810, 11, 12hlatjcl 32364 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
3938adantr 463 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
4010, 16, 17, 12meetat2 32295 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  A )  ->  (
( ( P  .\/  Q )  ./\  S )  e.  A  \/  (
( P  .\/  Q
)  ./\  S )  =  .0.  ) )
417, 39, 9, 40syl3anc 1230 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  S )  e.  A  \/  (
( P  .\/  Q
)  ./\  S )  =  .0.  ) )
4241adantr 463 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( ( P 
.\/  Q )  ./\  S )  e.  A  \/  ( ( P  .\/  Q )  ./\  S )  =  .0.  ) )
43 oveq1 6284 . . . . . . . . . . 11  |-  ( R  =  S  ->  ( R  .\/  S )  =  ( S  .\/  S
) )
4411, 12hlatjidm 32366 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
455, 9, 44syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( S  .\/  S )  =  S )
4643, 45sylan9eqr 2465 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( R  .\/  S
)  =  S )
4746oveq2d 6293 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  ./\  S )
)
4847eleq1d 2471 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  <->  ( ( P  .\/  Q )  ./\  S )  e.  A ) )
4947eqeq1d 2404 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  <->  ( ( P  .\/  Q )  ./\  S )  =  .0.  )
)
5048, 49orbi12d 708 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )  <->  ( (
( P  .\/  Q
)  ./\  S )  e.  A  \/  (
( P  .\/  Q
)  ./\  S )  =  .0.  ) ) )
5142, 50mpbird 232 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  R  =  S )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
5251adantlr 713 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  /\  R  =  S )  ->  ( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
53 df-ne 2600 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =/= 
.0. 
<->  -.  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )
54 simpll1 1036 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  K  e.  HL )
55 simpll2 1037 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  P  e.  A )
56 simpll3 1038 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  Q  e.  A )
57 simpr1 1003 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  P  =/=  Q )
58 eqid 2402 . . . . . . . . . . . . 13  |-  ( LLines `  K )  =  (
LLines `  K )
5911, 12, 58llni2 32509 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
6054, 55, 56, 57, 59syl31anc 1233 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  -> 
( P  .\/  Q
)  e.  ( LLines `  K ) )
61 simplr1 1039 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  R  e.  A )
62 simplr2 1040 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  S  e.  A )
63 simpr2 1004 . . . . . . . . . . . 12  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  ->  R  =/=  S )
6411, 12, 58llni2 32509 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
6554, 61, 62, 63, 64syl31anc 1233 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  -> 
( R  .\/  S
)  e.  ( LLines `  K ) )
66 simplr3 1041 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  -> 
( P  .\/  Q
)  =/=  ( R 
.\/  S ) )
67 simpr3 1005 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  -> 
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  )
6816, 17, 12, 582llnmat 32521 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  .\/  Q
)  e.  ( LLines `  K )  /\  ( R  .\/  S )  e.  ( LLines `  K )
)  /\  ( ( P  .\/  Q )  =/=  ( R  .\/  S
)  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/=  .0.  )
)  ->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
6954, 60, 65, 66, 67, 68syl32anc 1238 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  ( P  =/=  Q  /\  R  =/=  S  /\  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =/= 
.0.  ) )  -> 
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A )
70693exp2 1215 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q
)  =/=  ( R 
.\/  S ) ) )  ->  ( P  =/=  Q  ->  ( R  =/=  S  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =/= 
.0.  ->  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A ) ) ) )
7170imp31 430 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  /\  R  =/=  S )  -> 
( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =/=  .0.  ->  ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A ) )
7253, 71syl5bir 218 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  /\  R  =/=  S )  -> 
( -.  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0. 
->  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A ) )
7372orrd 376 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  /\  R  =/=  S )  -> 
( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A ) )
7473orcomd 386 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  /\  R  =/=  S )  -> 
( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
7552, 74pm2.61dane 2721 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  S  e.  A  /\  ( P  .\/  Q )  =/=  ( R  .\/  S ) ) )  /\  P  =/=  Q )  -> 
( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  ) )
7637, 75pm2.61dane 2721 . . 3  |-  ( ( ( 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.  ) )
771, 2, 3, 4, 76syl13anc 1232 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  e.  A )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )
78 simpl1 1000 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  HL )
7978, 6syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  K  e.  OL )
8038adantr 463 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
81 simpr1 1003 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  R  e.  A )
8210, 16, 17, 12meetat2 32295 . . . . 5  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  R  e.  A )  ->  (
( ( P  .\/  Q )  ./\  R )  e.  A  \/  (
( P  .\/  Q
)  ./\  R )  =  .0.  ) )
8379, 80, 81, 82syl3anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  R )  e.  A  \/  (
( P  .\/  Q
)  ./\  R )  =  .0.  ) )
8483adantr 463 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( ( P  .\/  Q )  ./\  R )  e.  A  \/  (
( P  .\/  Q
)  ./\  R )  =  .0.  ) )
85 oveq2 6285 . . . . . . 7  |-  ( S  =  .0.  ->  ( R  .\/  S )  =  ( R  .\/  .0.  ) )
8610, 12atbase 32287 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
8781, 86syl 17 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  R  e.  ( Base `  K )
)
8810, 11, 17olj01 32223 . . . . . . . 8  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
8979, 87, 88syl2anc 659 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( R  .\/  .0.  )  =  R )
9085, 89sylan9eqr 2465 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  ( R  .\/  S )  =  R )
9190oveq2d 6293 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =  ( ( P  .\/  Q )  ./\  R )
)
9291eleq1d 2471 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  <->  ( ( P 
.\/  Q )  ./\  R )  e.  A ) )
9391eqeq1d 2404 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  <->  ( ( P 
.\/  Q )  ./\  R )  =  .0.  )
)
9492, 93orbi12d 708 . . 3  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )  <->  ( (
( P  .\/  Q
)  ./\  R )  e.  A  \/  (
( P  .\/  Q
)  ./\  R )  =  .0.  ) ) )
9584, 94mpbird 232 . 2  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  /\  S  =  .0.  )  ->  (
( ( P  .\/  Q )  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )
96 simpr2 1004 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( S  e.  A  \/  S  =  .0.  ) )
9777, 95, 96mpjaodan 787 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( R  e.  A  /\  ( S  e.  A  \/  S  =  .0.  )  /\  ( P  .\/  Q )  =/=  ( R 
.\/  S ) ) )  ->  ( (
( P  .\/  Q
)  ./\  ( R  .\/  S ) )  e.  A  \/  ( ( P  .\/  Q ) 
./\  ( R  .\/  S ) )  =  .0.  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 366    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   ` cfv 5568  (class class class)co 6277   Basecbs 14839   joincjn 15895   meetcmee 15896   0.cp0 15989   Latclat 15997   OLcol 32172   Atomscatm 32261   HLchlt 32348   LLinesclln 32488
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
This theorem is referenced by:  2atmat0  32523  cdlemg31b0a  33694
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