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Theorem ps-1 28817
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 5785 . . . . . 6  |-  ( R  =  P  ->  ( R  .\/  S )  =  ( P  .\/  S
) )
21breq2d 3995 . . . . 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 28708 . . . . . . . . . . . 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 28704 . . . . . . . . . . . . . . . 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 28630 . . . . . . . . . . . . . . . 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 28630 . . . . . . . . . . . . . . . 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 28707 . . . . . . . . . . . . . . . 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 14116 . . . . . . . . . . . . . . 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 28733 . . . . . . . . . . . . 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 4009 . . . . . . . 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 28707 . . . . . . . . . . 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 14116 . . . . . . . . . 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 28733 . . . . . . . . . . 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 28707 . . . . . . 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 14116 . . . . . 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 28733 . . . . 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 28707 . . . . 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 14107 . . . 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 3987 . . 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 3983   ` cfv 4659  (class class class)co 5778   Basecbs 13096   lecple 13163   joincjn 14026   Latclat 14099   Atomscatm 28604   HLchlt 28691
This theorem is referenced by:  2atjlej  28819  hlatexch3N  28820  hlatexch4  28821  2llnjaN  28906  dalem1  28999  lneq2at  29118  2llnma3r  29128  cdleme11c  29601  cdleme11  29610  cdleme35a  29788  cdleme42k  29824  cdlemg8b  29968  cdlemg13a  29991  cdlemg18b  30019  cdlemg42  30069  trljco  30080
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 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-undef 6250  df-riota 6258  df-poset 14028  df-plt 14040  df-lub 14056  df-join 14058  df-lat 14100  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692
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