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Theorem ps-1 32843
Description: The join of two atoms  R  .\/  S (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)
Hypotheses
Ref Expression
ps1.l  |-  .<_  =  ( le `  K )
ps1.j  |-  .\/  =  ( join `  K )
ps1.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
ps-1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R  .\/  S ) ) )

Proof of Theorem ps-1
StepHypRef Expression
1 oveq1 6097 . . . . . 6  |-  ( R  =  P  ->  ( R  .\/  S )  =  ( P  .\/  S
) )
21breq2d 4301 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
31eqeq2d 2452 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  =  ( R 
.\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
42, 3imbi12d 320 . . . 4  |-  ( R  =  P  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
54eqcoms 2444 . . 3  |-  ( P  =  R  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
6 simp3 985 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  Q )  .<_  ( R 
.\/  S ) )
7 simp1 983 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
8 simp21 1016 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
9 simp3l 1011 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  R  e.  A )
10 ps1.j . . . . . . . . . . . . 13  |-  .\/  =  ( join `  K )
11 ps1.a . . . . . . . . . . . . 13  |-  A  =  ( Atoms `  K )
1210, 11hlatjcom 32734 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
137, 8, 9, 12syl3anc 1213 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  R
)  =  ( R 
.\/  P ) )
14133ad2ant1 1004 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  R )  =  ( R  .\/  P ) )
15 hllat 32730 . . . . . . . . . . . . . . . 16  |-  ( K  e.  HL  ->  K  e.  Lat )
16153ad2ant1 1004 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
17 eqid 2441 . . . . . . . . . . . . . . . . 17  |-  ( Base `  K )  =  (
Base `  K )
1817, 11atbase 32656 . . . . . . . . . . . . . . . 16  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
198, 18syl 16 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  ( Base `  K ) )
20 simp22 1017 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
2117, 11atbase 32656 . . . . . . . . . . . . . . . 16  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2220, 21syl 16 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  ( Base `  K ) )
23 simp3r 1012 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
2417, 10, 11hlatjcl 32733 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
257, 9, 23, 24syl3anc 1213 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( R  .\/  S
)  e.  ( Base `  K ) )
26 ps1.l . . . . . . . . . . . . . . . 16  |-  .<_  =  ( le `  K )
2717, 26, 10latjle12 15228 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  ( R  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( R  .\/  S )  /\  Q  .<_  ( R 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
2816, 19, 22, 25, 27syl13anc 1215 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .<_  ( R  .\/  S )  /\  Q  .<_  ( R 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
29 simpl 454 . . . . . . . . . . . . . 14  |-  ( ( P  .<_  ( R  .\/  S )  /\  Q  .<_  ( R  .\/  S
) )  ->  P  .<_  ( R  .\/  S
) )
3028, 29syl6bir 229 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  P  .<_  ( R  .\/  S
) ) )
3130adantr 462 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  P  .<_  ( R  .\/  S
) ) )
32 simpl1 986 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  ->  K  e.  HL )
33 simpl21 1061 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  ->  P  e.  A )
34 simpl3r 1039 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  ->  S  e.  A )
35 simpl3l 1038 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  ->  R  e.  A )
36 simpr 458 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  ->  P  =/=  R )
3726, 10, 11hlatexchb1 32759 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  R  e.  A
)  /\  P  =/=  R )  ->  ( P  .<_  ( R  .\/  S
)  <->  ( R  .\/  P )  =  ( R 
.\/  S ) ) )
3832, 33, 34, 35, 36, 37syl131anc 1226 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( P  .<_  ( R 
.\/  S )  <->  ( R  .\/  P )  =  ( R  .\/  S ) ) )
3931, 38sylibd 214 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( R  .\/  P )  =  ( R  .\/  S
) ) )
40393impia 1179 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( R  .\/  P )  =  ( R  .\/  S ) )
4114, 40eqtrd 2473 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  R )  =  ( R  .\/  S ) )
426, 41breqtrrd 4315 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  Q )  .<_  ( P 
.\/  R ) )
43423expia 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  .<_  ( P  .\/  R ) ) )
4417, 10, 11hlatjcl 32733 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
457, 8, 9, 44syl3anc 1213 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
4617, 26, 10latjle12 15228 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P 
.\/  R ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  R ) ) )
4716, 19, 22, 45, 46syl13anc 1215 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P 
.\/  R ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  R ) ) )
48 simpr 458 . . . . . . . . . 10  |-  ( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P  .\/  R
) )  ->  Q  .<_  ( P  .\/  R
) )
49 simp23 1018 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  =/=  Q )
5049necomd 2693 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  =/=  P )
5126, 10, 11hlatexchb1 32759 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  Q  =/=  P )  ->  ( Q  .<_  ( P  .\/  R
)  <->  ( P  .\/  Q )  =  ( P 
.\/  R ) ) )
527, 20, 9, 8, 50, 51syl131anc 1226 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( Q  .<_  ( P 
.\/  R )  <->  ( P  .\/  Q )  =  ( P  .\/  R ) ) )
5348, 52syl5ib 219 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P 
.\/  R ) )  ->  ( P  .\/  Q )  =  ( P 
.\/  R ) ) )
5447, 53sylbird 235 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( P  .\/  R )  ->  ( P  .\/  Q )  =  ( P  .\/  R
) ) )
5554adantr 462 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( P  .\/  R )  ->  ( P  .\/  Q )  =  ( P  .\/  R
) ) )
5643, 55syld 44 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( P  .\/  R
) ) )
57563impia 1179 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  Q )  =  ( P  .\/  R ) )
5857, 41eqtrd 2473 . . . 4  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R  /\  ( P  .\/  Q )  .<_  ( R  .\/  S ) )  ->  ( P  .\/  Q )  =  ( R  .\/  S ) )
59583expia 1184 . . 3  |-  ( ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  /\  P  =/=  R )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) ) )
6017, 10, 11hlatjcl 32733 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
617, 8, 23, 60syl3anc 1213 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  S
)  e.  ( Base `  K ) )
6217, 26, 10latjle12 15228 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
6316, 19, 22, 61, 62syl13anc 1215 . . . . 5  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P 
.\/  S ) )  <-> 
( P  .\/  Q
)  .<_  ( P  .\/  S ) ) )
64 simpr 458 . . . . 5  |-  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S
) )  ->  Q  .<_  ( P  .\/  S
) )
6563, 64syl6bir 229 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  Q  .<_  ( P  .\/  S
) ) )
6626, 10, 11hlatexchb1 32759 . . . . 5  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  S  e.  A  /\  P  e.  A
)  /\  Q  =/=  P )  ->  ( Q  .<_  ( P  .\/  S
)  <->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) )
677, 20, 23, 8, 50, 66syl131anc 1226 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( Q  .<_  ( P 
.\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
6865, 67sylibd 214 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P  .\/  S
) ) )
695, 59, 68pm2.61ne 2684 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) ) )
7017, 10, 11hlatjcl 32733 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
717, 8, 20, 70syl3anc 1213 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
7217, 26latref 15219 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  .<_  ( P  .\/  Q ) )
7316, 71, 72syl2anc 656 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( P  .\/  Q ) )
74 breq2 4293 . . 3  |-  ( ( P  .\/  Q )  =  ( R  .\/  S )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  Q )  <-> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
7573, 74syl5ibcom 220 . 2  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  =  ( R 
.\/  S )  -> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
7669, 75impbid 191 1  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( ( P  .\/  Q )  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  =  ( R  .\/  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   lecple 14241   joincjn 15110   Latclat 15211   Atomscatm 32630   HLchlt 32717
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-covers 32633  df-ats 32634  df-atl 32665  df-cvlat 32689  df-hlat 32718
This theorem is referenced by:  2atjlej  32845  hlatexch3N  32846  hlatexch4  32847  2llnjaN  32932  dalem1  33025  lneq2at  33144  2llnma3r  33154  cdleme11c  33627  cdleme11  33636  cdleme35a  33814  cdleme42k  33850  cdlemg8b  33994  cdlemg13a  34017  cdlemg18b  34045  cdlemg42  34095  trljco  34106
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