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Theorem cdleme9 29692
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114.  C and  F represent s1 and f(s) respectively. In their notation, we prove f(s)  \/ s1 = q  \/ s1. (Contributed by NM, 10-Jun-2012.)
Hypotheses
Ref Expression
cdleme9.l  |-  .<_  =  ( le `  K )
cdleme9.j  |-  .\/  =  ( join `  K )
cdleme9.m  |-  ./\  =  ( meet `  K )
cdleme9.a  |-  A  =  ( Atoms `  K )
cdleme9.h  |-  H  =  ( LHyp `  K
)
cdleme9.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme9.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
cdleme9.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
Assertion
Ref Expression
cdleme9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  C )  =  ( Q  .\/  C
) )

Proof of Theorem cdleme9
StepHypRef Expression
1 cdleme9.l . . . 4  |-  .<_  =  ( le `  K )
2 cdleme9.j . . . 4  |-  .\/  =  ( join `  K )
3 cdleme9.m . . . 4  |-  ./\  =  ( meet `  K )
4 cdleme9.a . . . 4  |-  A  =  ( Atoms `  K )
5 cdleme9.h . . . 4  |-  H  =  ( LHyp `  K
)
6 cdleme9.u . . . 4  |-  U  =  ( ( P  .\/  Q )  ./\  W )
7 cdleme9.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
8 cdleme9.c . . . 4  |-  C  =  ( ( P  .\/  S )  ./\  W )
91, 2, 3, 4, 5, 6, 7, 8cdleme3d 29670 . . 3  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  C ) )
109oveq1i 5802 . 2  |-  ( F 
.\/  C )  =  ( ( ( S 
.\/  U )  ./\  ( Q  .\/  C ) )  .\/  C )
11 simp1l 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  K  e.  HL )
12 simp1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp21 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
14 simp23l 1081 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  e.  A )
15 hllat 28803 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1611, 15syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  K  e.  Lat )
17 eqid 2258 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
1817, 4atbase 28729 . . . . . . 7  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1914, 18syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  e.  ( Base `  K
) )
20 simp21l 1077 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  e.  A )
2117, 4atbase 28729 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2220, 21syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  e.  ( Base `  K
) )
23 simp22 994 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  Q  e.  A )
2417, 4atbase 28729 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
2523, 24syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  Q  e.  ( Base `  K
) )
26 simp3 962 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  -.  S  .<_  ( P  .\/  Q ) )
2717, 1, 2latnlej1l 14138 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  S  =/=  P )
2827necomd 2504 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  S  .<_  ( P  .\/  Q ) )  ->  P  =/=  S )
2916, 19, 22, 25, 26, 28syl131anc 1200 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  =/=  S )
301, 2, 3, 4, 5, 8cdleme9a 29690 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( S  e.  A  /\  P  =/=  S ) )  ->  C  e.  A
)
3112, 13, 14, 29, 30syl112anc 1191 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  C  e.  A )
321, 2, 3, 4, 5, 6, 17cdleme0aa 29649 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
3312, 20, 23, 32syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  U  e.  ( Base `  K
) )
3417, 2latjcl 14119 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
3516, 19, 33, 34syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
3617, 2, 4hlatjcl 28806 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  C  e.  A )  ->  ( Q  .\/  C
)  e.  ( Base `  K ) )
3711, 23, 31, 36syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  C )  e.  ( Base `  K
) )
381, 2, 4hlatlej2 28815 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  C  e.  A )  ->  C  .<_  ( Q  .\/  C ) )
3911, 23, 31, 38syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  C  .<_  ( Q  .\/  C
) )
4017, 1, 2, 3, 4atmod4i1 29305 . . . 4  |-  ( ( K  e.  HL  /\  ( C  e.  A  /\  ( S  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  C )  e.  ( Base `  K
) )  /\  C  .<_  ( Q  .\/  C
) )  ->  (
( ( S  .\/  U )  ./\  ( Q  .\/  C ) )  .\/  C )  =  ( ( ( S  .\/  U
)  .\/  C )  ./\  ( Q  .\/  C
) ) )
4111, 31, 35, 37, 39, 40syl131anc 1200 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( ( S  .\/  U )  ./\  ( Q  .\/  C ) )  .\/  C )  =  ( ( ( S  .\/  U
)  .\/  C )  ./\  ( Q  .\/  C
) ) )
4217, 2, 4hlatjcl 28806 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
4311, 20, 14, 42syl3anc 1187 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
44 simp1r 985 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  W  e.  H )
4517, 5lhpbase 29437 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
4644, 45syl 17 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  W  e.  ( Base `  K
) )
471, 2, 4hlatlej2 28815 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
4811, 20, 14, 47syl3anc 1187 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  S  .<_  ( P  .\/  S
) )
4917, 1, 2, 3, 4atmod3i1 29303 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  S  .<_  ( P  .\/  S
) )  ->  ( S  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( S  .\/  W ) ) )
5011, 14, 43, 46, 48, 49syl131anc 1200 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  .\/  ( ( P 
.\/  S )  ./\  W ) )  =  ( ( P  .\/  S
)  ./\  ( S  .\/  W ) ) )
51 simp23r 1082 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  -.  S  .<_  W )
52 eqid 2258 . . . . . . . . . . 11  |-  ( 1.
`  K )  =  ( 1. `  K
)
531, 2, 52, 4, 5lhpjat2 29460 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  -> 
( S  .\/  W
)  =  ( 1.
`  K ) )
5412, 14, 51, 53syl12anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  .\/  W )  =  ( 1. `  K
) )
5554oveq2d 5808 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  S
)  ./\  ( S  .\/  W ) )  =  ( ( P  .\/  S )  ./\  ( 1. `  K ) ) )
56 hlol 28801 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
5711, 56syl 17 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  K  e.  OL )
5817, 3, 52olm11 28667 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
5957, 43, 58syl2anc 645 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  S
)  ./\  ( 1. `  K ) )  =  ( P  .\/  S
) )
6050, 55, 593eqtrrd 2295 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  S )  =  ( S  .\/  (
( P  .\/  S
)  ./\  W )
) )
618oveq2i 5803 . . . . . . 7  |-  ( S 
.\/  C )  =  ( S  .\/  (
( P  .\/  S
)  ./\  W )
)
6260, 61syl6reqr 2309 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( S  .\/  C )  =  ( P  .\/  S
) )
6362oveq1d 5807 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( S  .\/  C
)  .\/  U )  =  ( ( P 
.\/  S )  .\/  U ) )
6417, 4atbase 28729 . . . . . . 7  |-  ( C  e.  A  ->  C  e.  ( Base `  K
) )
6531, 64syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  C  e.  ( Base `  K
) )
6617, 2latj32 14166 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  C  e.  ( Base `  K
) ) )  -> 
( ( S  .\/  U )  .\/  C )  =  ( ( S 
.\/  C )  .\/  U ) )
6716, 19, 33, 65, 66syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( S  .\/  U
)  .\/  C )  =  ( ( S 
.\/  C )  .\/  U ) )
682, 4hlatj32 28811 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  Q  e.  A
) )  ->  (
( P  .\/  S
)  .\/  Q )  =  ( ( P 
.\/  Q )  .\/  S ) )
6911, 20, 14, 23, 68syl13anc 1189 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  S
)  .\/  Q )  =  ( ( P 
.\/  Q )  .\/  S ) )
7017, 2latjcom 14128 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  =  ( ( P  .\/  S
)  .\/  Q )
)
7116, 25, 43, 70syl3anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( P  .\/  S ) )  =  ( ( P  .\/  S
)  .\/  Q )
)
726oveq2i 5803 . . . . . . . . 9  |-  ( P 
.\/  U )  =  ( P  .\/  (
( P  .\/  Q
)  ./\  W )
)
7317, 2, 4hlatjcl 28806 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
7411, 20, 23, 73syl3anc 1187 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
751, 2, 4hlatlej1 28814 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
7611, 20, 23, 75syl3anc 1187 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  P  .<_  ( P  .\/  Q
) )
7717, 1, 2, 3, 4atmod3i1 29303 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  Q
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( P  .\/  W ) ) )
7811, 20, 74, 46, 76, 77syl131anc 1200 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( ( P  .\/  Q
)  ./\  ( P  .\/  W ) ) )
791, 2, 52, 4, 5lhpjat2 29460 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  .\/  W
)  =  ( 1.
`  K ) )
8012, 13, 79syl2anc 645 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  W )  =  ( 1. `  K
) )
8180oveq2d 5808 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( P  .\/  W ) )  =  ( ( P  .\/  Q )  ./\  ( 1. `  K ) ) )
8217, 3, 52olm11 28667 . . . . . . . . . . 11  |-  ( ( K  e.  OL  /\  ( P  .\/  Q )  e.  ( Base `  K
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
8357, 74, 82syl2anc 645 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  Q
)  ./\  ( 1. `  K ) )  =  ( P  .\/  Q
) )
8478, 81, 833eqtrd 2294 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  ( ( P 
.\/  Q )  ./\  W ) )  =  ( P  .\/  Q ) )
8572, 84syl5eq 2302 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( P  .\/  U )  =  ( P  .\/  Q
) )
8685oveq1d 5807 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  U
)  .\/  S )  =  ( ( P 
.\/  Q )  .\/  S ) )
8769, 71, 863eqtr4d 2300 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( P  .\/  S ) )  =  ( ( P  .\/  U
)  .\/  S )
)
8817, 2latj32 14166 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( ( P  .\/  U )  .\/  S )  =  ( ( P 
.\/  S )  .\/  U ) )
8916, 22, 33, 19, 88syl13anc 1189 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  U
)  .\/  S )  =  ( ( P 
.\/  S )  .\/  U ) )
9087, 89eqtrd 2290 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( P  .\/  S ) )  =  ( ( P  .\/  S
)  .\/  U )
)
9163, 67, 903eqtr4d 2300 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( S  .\/  U
)  .\/  C )  =  ( Q  .\/  ( P  .\/  S ) ) )
9291oveq1d 5807 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( ( S  .\/  U )  .\/  C ) 
./\  ( Q  .\/  C ) )  =  ( ( Q  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  C ) ) )
9317, 1, 3latmle1 14145 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
9416, 43, 46, 93syl3anc 1187 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( P  .\/  S
)  ./\  W )  .<_  ( P  .\/  S
) )
958, 94syl5eqbr 4030 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  C  .<_  ( P  .\/  S
) )
9617, 1, 2latjlej2 14135 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( C  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
) )  ->  ( C  .<_  ( P  .\/  S )  ->  ( Q  .\/  C )  .<_  ( Q 
.\/  ( P  .\/  S ) ) ) )
9716, 65, 43, 25, 96syl13anc 1189 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( C  .<_  ( P  .\/  S )  ->  ( Q  .\/  C )  .<_  ( Q 
.\/  ( P  .\/  S ) ) ) )
9895, 97mpd 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  C )  .<_  ( Q  .\/  ( P 
.\/  S ) ) )
9917, 2latjcl 14119 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
10016, 25, 43, 99syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
10117, 1, 3latleeqm2 14149 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  C )  e.  ( Base `  K
)  /\  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  C )  .<_  ( Q  .\/  ( P 
.\/  S ) )  <-> 
( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  C ) )  =  ( Q  .\/  C
) ) )
10216, 37, 100, 101syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  C
)  .<_  ( Q  .\/  ( P  .\/  S ) )  <->  ( ( Q 
.\/  ( P  .\/  S ) )  ./\  ( Q  .\/  C ) )  =  ( Q  .\/  C ) ) )
10398, 102mpbid 203 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( Q  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  C ) )  =  ( Q  .\/  C ) )
10441, 92, 1033eqtrd 2294 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  (
( ( S  .\/  U )  ./\  ( Q  .\/  C ) )  .\/  C )  =  ( Q 
.\/  C ) )
10510, 104syl5eq 2302 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( S  e.  A  /\  -.  S  .<_  W ) )  /\  -.  S  .<_  ( P  .\/  Q
) )  ->  ( F  .\/  C )  =  ( Q  .\/  C
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   joincjn 14041   meetcmee 14042   1.cp1 14107   Latclat 14114   OLcol 28614   Atomscatm 28703   HLchlt 28790   LHypclh 29423
This theorem is referenced by:  cdleme9tN  29696  cdleme17a  29725
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 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791  df-psubsp 28942  df-pmap 28943  df-padd 29235  df-lhyp 29427
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