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Theorem ps-1 28796
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 5764 . . . . . 6  |-  ( R  =  P  ->  ( R  .\/  S )  =  ( P  .\/  S
) )
21breq2d 3975 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
31eqeq2d 2267 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  =  ( R 
.\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
42, 3imbi12d 313 . . . 4  |-  ( R  =  P  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
54eqcoms 2259 . . 3  |-  ( P  =  R  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
6 simp3 962 . . . . . . . . 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 960 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
8 simp21 993 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
9 simp3l 988 . . . . . . . . . . . 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 28687 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
137, 8, 9, 12syl3anc 1187 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  R
)  =  ( R 
.\/  P ) )
14133ad2ant1 981 . . . . . . . . . 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 28683 . . . . . . . . . . . . . . . 16  |-  ( K  e.  HL  ->  K  e.  Lat )
16153ad2ant1 981 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
17 eqid 2256 . . . . . . . . . . . . . . . . 17  |-  ( Base `  K )  =  (
Base `  K )
1817, 11atbase 28609 . . . . . . . . . . . . . . . 16  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
198, 18syl 17 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  ( Base `  K ) )
20 simp22 994 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
2117, 11atbase 28609 . . . . . . . . . . . . . . . 16  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2220, 21syl 17 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  ( Base `  K ) )
23 simp3r 989 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
2417, 10, 11hlatjcl 28686 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
257, 9, 23, 24syl3anc 1187 . . . . . . . . . . . . . . 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 14095 . . . . . . . . . . . . . . 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 1189 . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . 14  |-  ( ( P  .<_  ( R  .\/  S )  /\  Q  .<_  ( R  .\/  S
) )  ->  P  .<_  ( R  .\/  S
) )
3028, 29syl6bir 222 . . . . . . . . . . . . 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 453 . . . . . . . . . . . 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 963 . . . . . . . . . . . . 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 1038 . . . . . . . . . . . . 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 1016 . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 28712 . . . . . . . . . . . . 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 1200 . . . . . . . . . . . 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 207 . . . . . . . . . . 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 1153 . . . . . . . . . 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 2288 . . . . . . . . 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 3989 . . . . . . . 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 1158 . . . . . . 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 28686 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
457, 8, 9, 44syl3anc 1187 . . . . . . . . . 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 14095 . . . . . . . . . 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 1189 . . . . . . . . 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 449 . . . . . . . . . 10  |-  ( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P  .\/  R
) )  ->  Q  .<_  ( P  .\/  R
) )
49 simp23 995 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  =/=  Q )
5049necomd 2502 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  =/=  P )
5126, 10, 11hlatexchb1 28712 . . . . . . . . . . 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 1200 . . . . . . . . . 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 212 . . . . . . . . 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 228 . . . . . . . 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 453 . . . . . . 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 42 . . . . . 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 1153 . . . . 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 2288 . . . 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 1158 . . 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 28686 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
617, 8, 23, 60syl3anc 1187 . . . . . 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 14095 . . . . . 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 1189 . . . . 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 449 . . . . 5  |-  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S
) )  ->  Q  .<_  ( P  .\/  S
) )
6563, 64syl6bir 222 . . . 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 28712 . . . . 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 1200 . . . 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 207 . . 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 2494 . 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 28686 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
717, 8, 20, 70syl3anc 1187 . . . 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 14086 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  .<_  ( P  .\/  Q ) )
7316, 71, 72syl2anc 645 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( P  .\/  Q ) )
74 breq2 3967 . . 3  |-  ( ( P  .\/  Q )  =  ( R  .\/  S )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  Q )  <-> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
7573, 74syl5ibcom 213 . 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 185 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 set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Basecbs 13075   lecple 13142   joincjn 14005   Latclat 14078   Atomscatm 28583   HLchlt 28670
This theorem is referenced by:  2atjlej  28798  hlatexch3N  28799  hlatexch4  28800  2llnjaN  28885  dalem1  28978  lneq2at  29097  2llnma3r  29107  cdleme11c  29580  cdleme11  29589  cdleme35a  29767  cdleme42k  29803  cdlemg8b  29947  cdlemg13a  29970  cdlemg18b  29998  cdlemg42  30048  trljco  30059
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-undef 6229  df-riota 6237  df-poset 14007  df-plt 14019  df-lub 14035  df-join 14037  df-lat 14079  df-covers 28586  df-ats 28587  df-atl 28618  df-cvlat 28642  df-hlat 28671
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