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Theorem ps-1 30288
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 5881 . . . . . 6  |-  ( R  =  P  ->  ( R  .\/  S )  =  ( P  .\/  S
) )
21breq2d 4051 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
31eqeq2d 2307 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  =  ( R 
.\/  S )  <->  ( P  .\/  Q )  =  ( P  .\/  S ) ) )
42, 3imbi12d 311 . . . 4  |-  ( R  =  P  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
54eqcoms 2299 . . 3  |-  ( P  =  R  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
6 simp3 957 . . . . . . . . 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 955 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
8 simp21 988 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
9 simp3l 983 . . . . . . . . . . . 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 30179 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
137, 8, 9, 12syl3anc 1182 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  R
)  =  ( R 
.\/  P ) )
14133ad2ant1 976 . . . . . . . . . 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 30175 . . . . . . . . . . . . . . . 16  |-  ( K  e.  HL  ->  K  e.  Lat )
16153ad2ant1 976 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
17 eqid 2296 . . . . . . . . . . . . . . . . 17  |-  ( Base `  K )  =  (
Base `  K )
1817, 11atbase 30101 . . . . . . . . . . . . . . . 16  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
198, 18syl 15 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  ( Base `  K ) )
20 simp22 989 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
2117, 11atbase 30101 . . . . . . . . . . . . . . . 16  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2220, 21syl 15 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  ( Base `  K ) )
23 simp3r 984 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
2417, 10, 11hlatjcl 30178 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
257, 9, 23, 24syl3anc 1182 . . . . . . . . . . . . . . 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 14184 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . . 14  |-  ( ( P  .<_  ( R  .\/  S )  /\  Q  .<_  ( R  .\/  S
) )  ->  P  .<_  ( R  .\/  S
) )
3028, 29syl6bir 220 . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 958 . . . . . . . . . . . . 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 1033 . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . 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 447 . . . . . . . . . . . . 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 30204 . . . . . . . . . . . . 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 1195 . . . . . . . . . . . 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 205 . . . . . . . . . . 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 1148 . . . . . . . . . 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 2328 . . . . . . . . 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 4065 . . . . . . . 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 1153 . . . . . . 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 30178 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
457, 8, 9, 44syl3anc 1182 . . . . . . . . . 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 14184 . . . . . . . . . 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 1184 . . . . . . . . 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 447 . . . . . . . . . 10  |-  ( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P  .\/  R
) )  ->  Q  .<_  ( P  .\/  R
) )
49 simp23 990 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  =/=  Q )
5049necomd 2542 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  =/=  P )
5126, 10, 11hlatexchb1 30204 . . . . . . . . . . 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 1195 . . . . . . . . . 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 210 . . . . . . . . 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 226 . . . . . . . 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 451 . . . . . . 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 40 . . . . . 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 1148 . . . . 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 2328 . . . 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 1153 . . 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 30178 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
617, 8, 23, 60syl3anc 1182 . . . . . 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 14184 . . . . . 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 1184 . . . . 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 447 . . . . 5  |-  ( ( P  .<_  ( P  .\/  S )  /\  Q  .<_  ( P  .\/  S
) )  ->  Q  .<_  ( P  .\/  S
) )
6563, 64syl6bir 220 . . . 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 30204 . . . . 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 1195 . . . 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 205 . . 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 2534 . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
717, 8, 20, 70syl3anc 1182 . . . 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 14175 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  .<_  ( P  .\/  Q ) )
7316, 71, 72syl2anc 642 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( P  .\/  Q ) )
74 breq2 4043 . . 3  |-  ( ( P  .\/  Q )  =  ( R  .\/  S )  ->  ( ( P  .\/  Q )  .<_  ( P  .\/  Q )  <-> 
( P  .\/  Q
)  .<_  ( R  .\/  S ) ) )
7573, 74syl5ibcom 211 . 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 183 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  2atjlej  30290  hlatexch3N  30291  hlatexch4  30292  2llnjaN  30377  dalem1  30470  lneq2at  30589  2llnma3r  30599  cdleme11c  31072  cdleme11  31081  cdleme35a  31259  cdleme42k  31295  cdlemg8b  31439  cdlemg13a  31462  cdlemg18b  31490  cdlemg42  31540  trljco  31551
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-join 14126  df-lat 14168  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163
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