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Theorem 2at0mat0 30336
Description: Special case of 2atmat0 30337 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 730 . . 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 997 . . 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 447 . . 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 999 . . 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 958 . . . . . . . 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 30173 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  OL )
75, 6syl 15 . . . . . . 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 961 . . . . . . . 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 962 . . . . . . . 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 2296 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
11 2atmatz.j . . . . . . . . 9  |-  .\/  =  ( join `  K )
12 2atmatz.a . . . . . . . . 9  |-  A  =  ( Atoms `  K )
1310, 11, 12hlatjcl 30178 . . . . . . . 8  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
145, 8, 9, 13syl3anc 1182 . . . . . . 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 960 . . . . . . 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 30109 . . . . . . 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 1182 . . . . . 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 451 . . . . 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 5881 . . . . . . . . . 10  |-  ( P  =  Q  ->  ( P  .\/  Q )  =  ( Q  .\/  Q
) )
2211, 12hlatjidm 30180 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
235, 15, 22syl2anc 642 . . . . . . . . . 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 2350 . . . . . . . . 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 5889 . . . . . . . 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 30175 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
275, 26syl 15 . . . . . . . . . 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 30101 . . . . . . . . . . 11  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2915, 28syl 15 . . . . . . . . . 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 14197 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  ./\  ( R  .\/  S ) )  =  ( ( R  .\/  S
)  ./\  Q )
)
3127, 29, 14, 30syl3anc 1182 . . . . . . . . 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 451 . . . . . . . 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 2328 . . . . . . 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 2362 . . . . . 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 2304 . . . . . 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 690 . . . . 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 223 . . . 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 30178 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
3938adantr 451 . . . . . . . . 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 30109 . . . . . . . . 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 1182 . . . . . . . 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 451 . . . . . . 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 5881 . . . . . . . . . . 11  |-  ( R  =  S  ->  ( R  .\/  S )  =  ( S  .\/  S
) )
4411, 12hlatjidm 30180 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  S  e.  A )  ->  ( S  .\/  S
)  =  S )
455, 9, 44syl2anc 642 . . . . . . . . . . 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 2350 . . . . . . . . . 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 5890 . . . . . . . . 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 2362 . . . . . . . 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 2304 . . . . . . . 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 690 . . . . . . 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 223 . . . . . 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 695 . . . . 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 2461 . . . . . . . 8  |-  ( ( ( P  .\/  Q
)  ./\  ( R  .\/  S ) )  =/= 
.0. 
<->  -.  ( ( P 
.\/  Q )  ./\  ( R  .\/  S ) )  =  .0.  )
54 simpll1 994 . . . . . . . . . . 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 995 . . . . . . . . . . . 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 996 . . . . . . . . . . . 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 961 . . . . . . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( LLines `  K )  =  (
LLines `  K )
5911, 12, 58llni2 30323 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  Q )  e.  (
LLines `  K ) )
6054, 55, 56, 57, 59syl31anc 1185 . . . . . . . . . . 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 997 . . . . . . . . . . . 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 998 . . . . . . . . . . . 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 962 . . . . . . . . . . . 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 30323 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  /\  R  =/=  S
)  ->  ( R  .\/  S )  e.  (
LLines `  K ) )
6554, 61, 62, 63, 64syl31anc 1185 . . . . . . . . . . 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 999 . . . . . . . . . . 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 963 . . . . . . . . . . 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 30335 . . . . . . . . . . 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 1190 . . . . . . . . . 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 1169 . . . . . . . . 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 421 . . . . . . . 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 209 . . . . . . 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 367 . . . . . 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 377 . . . . 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 2537 . . . 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 2537 . . 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 1184 . 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 958 . . . . . 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 15 . . . . 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 451 . . . . 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 961 . . . . 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 30109 . . . . 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 1182 . . . 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 451 . . 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 5882 . . . . . . 7  |-  ( S  =  .0.  ->  ( R  .\/  S )  =  ( R  .\/  .0.  ) )
8610, 12atbase 30101 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
8781, 86syl 15 . . . . . . . 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 30037 . . . . . . . 8  |-  ( ( K  e.  OL  /\  R  e.  ( Base `  K ) )  -> 
( R  .\/  .0.  )  =  R )
8979, 87, 88syl2anc 642 . . . . . . 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 2350 . . . . . 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 5890 . . . . 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 2362 . . . 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 2304 . . . 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 690 . . 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 223 . 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 962 . 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 761 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   ` cfv 5271  (class class class)co 5874   Basecbs 13164   joincjn 14094   meetcmee 14095   0.cp0 14159   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162   LLinesclln 30302
This theorem is referenced by:  2atmat0  30337  cdlemg31b0a  31506
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
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