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Theorem 2at0mat0 32891
Description: Special case of 2atmat0 32892 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 748 . . 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 1025 . . 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 458 . . 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 1027 . . 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 986 . . . . . . . 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 32728 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
75, 6syl 16 . . . . . . 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 989 . . . . . . . 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 990 . . . . . . . 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 2441 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
11 2atmatz.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
12 2atmatz.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1310, 11, 12hlatjcl 32733 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
145, 8, 9, 13syl3anc 1213 . . . . . . 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 988 . . . . . . 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 32664 . . . . . . 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 1213 . . . . . 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 462 . . . . 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 6097 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2211, 12hlatjidm 32735 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
235, 15, 22syl2anc 656 . . . . . . . . . 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 2495 . . . . . . . . 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 6105 . . . . . . . 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 32730 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
275, 26syl 16 . . . . . . . . . 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 32656 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2915, 28syl 16 . . . . . . . . . 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 15241 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  S
)  ./\  Q )
)
3127, 29, 14, 30syl3anc 1213 . . . . . . . . 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 462 . . . . . . . 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 2473 . . . . . . 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 2507 . . . . . 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 2449 . . . . . 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 704 . . . . 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 32733 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
3938adantr 462 . . . . . . . . 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 32664 . . . . . . . . 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 1213 . . . . . . . 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 462 . . . . . . 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 6097 . . . . . . . . . . 11  |-  ( R  =  S  ->  ( R  .\/  S )  =  ( S  .\/  S
) )
4411, 12hlatjidm 32735 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
455, 9, 44syl2anc 656 . . . . . . . . . . 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 2495 . . . . . . . . . 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 6106 . . . . . . . . 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 2507 . . . . . . . 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 2449 . . . . . . . 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 704 . . . . . . 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 709 . . . . 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 2606 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =/= 
.0. 
<->  -.  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )
54 simpll1 1022 . . . . . . . . . . 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 1023 . . . . . . . . . . . 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 1024 . . . . . . . . . . . 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 989 . . . . . . . . . . . 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 2441 . . . . . . . . . . . . 13  |-  ( LLines `  K )  =  (
LLines `  K )
5911, 12, 58llni2 32878 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
6054, 55, 56, 57, 59syl31anc 1216 . . . . . . . . . . 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 1025 . . . . . . . . . . . 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 1026 . . . . . . . . . . . 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 990 . . . . . . . . . . . 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 32878 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
6554, 61, 62, 63, 64syl31anc 1216 . . . . . . . . . . 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 1027 . . . . . . . . . . 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 991 . . . . . . . . . . 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 32890 . . . . . . . . . . 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 1221 . . . . . . . . . 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 1200 . . . . . . . . 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 432 . . . . . . . 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 378 . . . . . 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 388 . . . . 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 2687 . . . 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 2687 . . 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 1215 . 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 986 . . . . . 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 16 . . . . 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 462 . . . . 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 989 . . . . 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 32664 . . . . 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 1213 . . . 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 462 . . 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 6098 . . . . . . 7  |-  ( S  =  .0.  ->  ( R  .\/  S )  =  ( R  .\/  .0.  ) )
8610, 12atbase 32656 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
8781, 86syl 16 . . . . . . . 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 32592 . . . . . . . 8  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
8979, 87, 88syl2anc 656 . . . . . . 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 2495 . . . . . 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 6106 . . . . 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 2507 . . . 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 2449 . . . 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 704 . . 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 990 . 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 779 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 368    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   ` cfv 5415  (class class class)co 6090   Basecbs 14170   joincjn 15110   meetcmee 15111   0.cp0 15203   Latclat 15211   OLcol 32541   Atomscatm 32630   HLchlt 32717   LLinesclln 32857
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-poset 15112  df-plt 15124  df-lub 15140  df-glb 15141  df-join 15142  df-meet 15143  df-p0 15205  df-lat 15212  df-clat 15274  df-oposet 32543  df-ol 32545  df-oml 32546  df-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718  df-llines 32864
This theorem is referenced by:  2atmat0  32892  cdlemg31b0a  34061
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