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Theorem cdlemf1 35232
Description: Part of Lemma F in [Crawley] p. 116. TODO: should this or part of it become a stand-alone theorem? (Contributed by NM, 12-Apr-2013.)
Hypotheses
Ref Expression
cdlemf1.l  |-  .<_  =  ( le `  K )
cdlemf1.j  |-  .\/  =  ( join `  K )
cdlemf1.a  |-  A  =  ( Atoms `  K )
cdlemf1.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdlemf1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( P  .\/  q ) ) )
Distinct variable groups:    A, q    H, q    K, q    .<_ , q    P, q    U, q    W, q
Allowed substitution hint:    .\/ ( q)

Proof of Theorem cdlemf1
StepHypRef Expression
1 simp1l 1015 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 simp3l 1019 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A
)
3 simp2l 1017 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  A
)
4 simp2r 1018 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  .<_  W )
5 simp3r 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
6 nbrne2 4458 . . . . 5  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
76necomd 2731 . . . 4  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  P  =/=  U
)
84, 5, 7syl2anc 661 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  U
)
9 cdlemf1.l . . . 4  |-  .<_  =  ( le `  K )
10 cdlemf1.j . . . 4  |-  .\/  =  ( join `  K )
11 cdlemf1.a . . . 4  |-  A  =  ( Atoms `  K )
129, 10, 11hlsupr 34057 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  U  e.  A )  /\  P  =/=  U
)  ->  E. q  e.  A  ( q  =/=  P  /\  q  =/= 
U  /\  q  .<_  ( P  .\/  U ) ) )
131, 2, 3, 8, 12syl31anc 1226 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P 
.\/  U ) ) )
14 simp31 1027 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  q  =/=  P )
1514necomd 2731 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  P  =/=  q )
16 simp13r 1107 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  -.  P  .<_  W )
17 simp12r 1105 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  U  .<_  W )
18 simp11l 1102 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  K  e.  HL )
19 hllat 34035 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
2018, 19syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  K  e.  Lat )
21 eqid 2460 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2221, 11atbase 33961 . . . . . . . . . . 11  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
23223ad2ant2 1013 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  q  e.  ( Base `  K
) )
24 simp12l 1104 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  U  e.  A )
2521, 11atbase 33961 . . . . . . . . . . 11  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2624, 25syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  U  e.  ( Base `  K
) )
27 simp11r 1103 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  W  e.  H )
28 cdlemf1.h . . . . . . . . . . . 12  |-  H  =  ( LHyp `  K
)
2921, 28lhpbase 34669 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3027, 29syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  W  e.  ( Base `  K
) )
3121, 9, 10latjle12 15538 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( q  e.  (
Base `  K )  /\  U  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( ( q 
.<_  W  /\  U  .<_  W )  <->  ( q  .\/  U )  .<_  W )
)
3220, 23, 26, 30, 31syl13anc 1225 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
( q  .<_  W  /\  U  .<_  W )  <->  ( q  .\/  U )  .<_  W ) )
3332biimpd 207 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
( q  .<_  W  /\  U  .<_  W )  -> 
( q  .\/  U
)  .<_  W ) )
3417, 33mpan2d 674 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
q  .<_  W  ->  (
q  .\/  U )  .<_  W ) )
35 simp33 1029 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  q  .<_  ( P  .\/  U
) )
36 hlcvl 34031 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  CvLat )
3718, 36syl 16 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  K  e.  CvLat )
38 simp2 992 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  q  e.  A )
39 simp13l 1106 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  P  e.  A )
40 simp32 1028 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  q  =/=  U )
419, 10, 11cvlatexch2 34009 . . . . . . . . . 10  |-  ( ( K  e.  CvLat  /\  (
q  e.  A  /\  P  e.  A  /\  U  e.  A )  /\  q  =/=  U
)  ->  ( q  .<_  ( P  .\/  U
)  ->  P  .<_  ( q  .\/  U ) ) )
4237, 38, 39, 24, 40, 41syl131anc 1236 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
q  .<_  ( P  .\/  U )  ->  P  .<_  ( q  .\/  U ) ) )
4335, 42mpd 15 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  P  .<_  ( q  .\/  U
) )
4421, 11atbase 33961 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4539, 44syl 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  P  e.  ( Base `  K
) )
4621, 10, 11hlatjcl 34038 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  q  e.  A  /\  U  e.  A )  ->  ( q  .\/  U
)  e.  ( Base `  K ) )
4718, 38, 24, 46syl3anc 1223 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
q  .\/  U )  e.  ( Base `  K
) )
4821, 9lattr 15532 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( q  .\/  U
)  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( q  .\/  U )  /\  ( q  .\/  U )  .<_  W )  ->  P  .<_  W )
)
4920, 45, 47, 30, 48syl13anc 1225 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
( P  .<_  ( q 
.\/  U )  /\  ( q  .\/  U
)  .<_  W )  ->  P  .<_  W ) )
5043, 49mpand 675 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
( q  .\/  U
)  .<_  W  ->  P  .<_  W ) )
5134, 50syld 44 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
q  .<_  W  ->  P  .<_  W ) )
5216, 51mtod 177 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  -.  q  .<_  W )
539, 10, 11cvlatexch1 34008 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  (
q  e.  A  /\  U  e.  A  /\  P  e.  A )  /\  q  =/=  P
)  ->  ( q  .<_  ( P  .\/  U
)  ->  U  .<_  ( P  .\/  q ) ) )
5437, 38, 24, 39, 14, 53syl131anc 1236 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  (
q  .<_  ( P  .\/  U )  ->  U  .<_  ( P  .\/  q ) ) )
5535, 54mpd 15 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  U  .<_  ( P  .\/  q
) )
5615, 52, 553jca 1171 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  /\  q  e.  A  /\  ( q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U ) ) )  ->  ( P  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( P  .\/  q
) ) )
57563exp 1190 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( q  e.  A  ->  ( (
q  =/=  P  /\  q  =/=  U  /\  q  .<_  ( P  .\/  U
) )  ->  ( P  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( P  .\/  q
) ) ) ) )
5857reximdvai 2928 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( E. q  e.  A  ( q  =/=  P  /\  q  =/= 
U  /\  q  .<_  ( P  .\/  U ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( P  .\/  q ) ) ) )
5913, 58mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  E. q  e.  A  ( P  =/=  q  /\  -.  q  .<_  W  /\  U  .<_  ( P  .\/  q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762    =/= wne 2655   E.wrex 2808   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   Basecbs 14479   lecple 14551   joincjn 15420   Latclat 15521   Atomscatm 33935   CvLatclc 33937   HLchlt 34022   LHypclh 34655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-lat 15522  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-lhyp 34659
This theorem is referenced by:  cdlemf2  35233  cdlemg5  35276
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