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Theorem 2at0mat0 35124
Description: Special case of 2atmat0 35125 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 753 . . 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 461 . . 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 34961 . . . . . . . 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 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 2443 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
11 2atmatz.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
12 2atmatz.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1310, 11, 12hlatjcl 34966 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
145, 8, 9, 13syl3anc 1229 . . . . . . 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 34897 . . . . . . 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 1229 . . . . . 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 465 . . . . 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 6288 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2211, 12hlatjidm 34968 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
235, 15, 22syl2anc 661 . . . . . . . . . 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 2506 . . . . . . . . 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 6296 . . . . . . . 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 34963 . . . . . . . . . . 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 34889 . . . . . . . . . . 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 15684 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  S
)  ./\  Q )
)
3127, 29, 14, 30syl3anc 1229 . . . . . . . . 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 465 . . . . . . . 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 2484 . . . . . . 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 2512 . . . . . 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 2445 . . . . . 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 709 . . . . 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 34966 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
3938adantr 465 . . . . . . . . 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 34897 . . . . . . . . 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 1229 . . . . . . . 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 465 . . . . . . 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 6288 . . . . . . . . . . 11  |-  ( R  =  S  ->  ( R  .\/  S )  =  ( S  .\/  S
) )
4411, 12hlatjidm 34968 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
455, 9, 44syl2anc 661 . . . . . . . . . . 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 2506 . . . . . . . . . 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 6297 . . . . . . . . 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 2512 . . . . . . . 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 2445 . . . . . . . 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 709 . . . . . . 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 714 . . . . 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 2640 . . . . . . . 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 2443 . . . . . . . . . . . . 13  |-  ( LLines `  K )  =  (
LLines `  K )
5911, 12, 58llni2 35111 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
6054, 55, 56, 57, 59syl31anc 1232 . . . . . . . . . . 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 35111 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
6554, 61, 62, 63, 64syl31anc 1232 . . . . . . . . . . 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 35123 . . . . . . . . . . 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 1237 . . . . . . . . . 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 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 2761 . . . 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 2761 . . 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 1231 . 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 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 465 . . . . 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 34897 . . . . 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 1229 . . . 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 465 . . 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 6289 . . . . . . 7  |-  ( S  =  .0.  ->  ( R  .\/  S )  =  ( R  .\/  .0.  ) )
8610, 12atbase 34889 . . . . . . . . 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 34825 . . . . . . . 8  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
8979, 87, 88syl2anc 661 . . . . . . 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 2506 . . . . . 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 6297 . . . . 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 2512 . . . 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 2445 . . . 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 709 . . 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 786 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   ` cfv 5578  (class class class)co 6281   Basecbs 14614   joincjn 15552   meetcmee 15553   0.cp0 15646   Latclat 15654   OLcol 34774   Atomscatm 34863   HLchlt 34950   LLinesclln 35090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-preset 15536  df-poset 15554  df-plt 15567  df-lub 15583  df-glb 15584  df-join 15585  df-meet 15586  df-p0 15648  df-lat 15655  df-clat 15717  df-oposet 34776  df-ol 34778  df-oml 34779  df-covers 34866  df-ats 34867  df-atl 34898  df-cvlat 34922  df-hlat 34951  df-llines 35097
This theorem is referenced by:  2atmat0  35125  cdlemg31b0a  36296
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