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Theorem ps-1 33126
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 6103 . . . . . 6  |-  ( R  =  P  ->  ( R  .\/  S )  =  ( P  .\/  S
) )
21breq2d 4309 . . . . 5  |-  ( R  =  P  ->  (
( P  .\/  Q
)  .<_  ( R  .\/  S )  <->  ( P  .\/  Q )  .<_  ( P  .\/  S ) ) )
31eqeq2d 2454 . . . . 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 2446 . . 3  |-  ( P  =  R  ->  (
( ( P  .\/  Q )  .<_  ( R  .\/  S )  ->  ( P  .\/  Q )  =  ( R  .\/  S
) )  <->  ( ( P  .\/  Q )  .<_  ( P  .\/  S )  ->  ( P  .\/  Q )  =  ( P 
.\/  S ) ) ) )
6 simp3 990 . . . . . . . . 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 988 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  HL )
8 simp21 1021 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  e.  A )
9 simp3l 1016 . . . . . . . . . . . 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 33017 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
137, 8, 9, 12syl3anc 1218 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  R
)  =  ( R 
.\/  P ) )
14133ad2ant1 1009 . . . . . . . . . 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 33013 . . . . . . . . . . . . . . . 16  |-  ( K  e.  HL  ->  K  e.  Lat )
16153ad2ant1 1009 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  K  e.  Lat )
17 eqid 2443 . . . . . . . . . . . . . . . . 17  |-  ( Base `  K )  =  (
Base `  K )
1817, 11atbase 32939 . . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  e.  A )
2117, 11atbase 32939 . . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  S  e.  A )
2417, 10, 11hlatjcl 33016 . . . . . . . . . . . . . . . 16  |-  ( ( K  e.  HL  /\  R  e.  A  /\  S  e.  A )  ->  ( R  .\/  S
)  e.  ( Base `  K ) )
257, 9, 23, 24syl3anc 1218 . . . . . . . . . . . . . . 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 15237 . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . 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 991 . . . . . . . . . . . . 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 1066 . . . . . . . . . . . . 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 1044 . . . . . . . . . . . . 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 1043 . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 33042 . . . . . . . . . . . . 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 1231 . . . . . . . . . . . 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 1184 . . . . . . . . . 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 2475 . . . . . . . . 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 4323 . . . . . . . 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 1189 . . . . . . 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 33016 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
457, 8, 9, 44syl3anc 1218 . . . . . . . . . 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 15237 . . . . . . . . . 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 1220 . . . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( P  .<_  ( P  .\/  R )  /\  Q  .<_  ( P  .\/  R
) )  ->  Q  .<_  ( P  .\/  R
) )
49 simp23 1023 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  P  =/=  Q )
5049necomd 2700 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  ->  Q  =/=  P )
5126, 10, 11hlatexchb1 33042 . . . . . . . . . . 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 1231 . . . . . . . . . 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 465 . . . . . . 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 1184 . . . . 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 2475 . . . 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 1189 . . 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 33016 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
617, 8, 23, 60syl3anc 1218 . . . . . 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 15237 . . . . . 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 1220 . . . . 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 461 . . . . 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 33042 . . . . 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 1231 . . . 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 2691 . 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 33016 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
717, 8, 20, 70syl3anc 1218 . . . 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 15228 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  .<_  ( P  .\/  Q ) )
7316, 71, 72syl2anc 661 . . 3  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  P  =/=  Q
)  /\  ( R  e.  A  /\  S  e.  A ) )  -> 
( P  .\/  Q
)  .<_  ( P  .\/  Q ) )
74 breq2 4301 . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   Latclat 15220   Atomscatm 32913   HLchlt 33000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-lat 15221  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001
This theorem is referenced by:  2atjlej  33128  hlatexch3N  33129  hlatexch4  33130  2llnjaN  33215  dalem1  33308  lneq2at  33427  2llnma3r  33437  cdleme11c  33910  cdleme11  33919  cdleme35a  34097  cdleme42k  34133  cdlemg8b  34277  cdlemg13a  34300  cdlemg18b  34328  cdlemg42  34378  trljco  34389
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