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Theorem cdlemf1 31197
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 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 simp3l 985 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A
)
3 simp2l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  A
)
4 simp2r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  .<_  W )
5 simp3r 986 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
6 nbrne2 4222 . . . . 5  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
76necomd 2681 . . . 4  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  P  =/=  U
)
84, 5, 7syl2anc 643 . . 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 30022 . . 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 1187 . 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 993 . . . . . 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 2681 . . . . 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 1073 . . . . . 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 1071 . . . . . . . 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 1068 . . . . . . . . . . 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 30000 . . . . . . . . . . 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 2435 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2221, 11atbase 29926 . . . . . . . . . . 11  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
23223ad2ant2 979 . . . . . . . . . 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 1070 . . . . . . . . . . 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 29926 . . . . . . . . . . 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 1069 . . . . . . . . . . 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 30634 . . . . . . . . . . 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 14479 . . . . . . . . . 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 1186 . . . . . . . . 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 199 . . . . . . . 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 656 . . . . . . 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 995 . . . . . . . . 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 29996 . . . . . . . . . . 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 958 . . . . . . . . . 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 1072 . . . . . . . . . 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 994 . . . . . . . . . 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 29974 . . . . . . . . . 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 1197 . . . . . . . . 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 29926 . . . . . . . . . 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 30003 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  q  e.  A  /\  U  e.  A )  ->  ( q  .\/  U
)  e.  ( Base `  K ) )
4718, 38, 24, 46syl3anc 1184 . . . . . . . . 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 14473 . . . . . . . . 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 1186 . . . . . . . 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 657 . . . . . . 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 42 . . . . . 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 170 . . . . 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 29973 . . . . . . 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 1197 . . . . . 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 1134 . . . 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 1152 . . 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 2808 . 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   Latclat 14462   Atomscatm 29900   CvLatclc 29902   HLchlt 29987   LHypclh 30620
This theorem is referenced by:  cdlemf2  31198  cdlemg5  31241
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-poset 14391  df-plt 14403  df-lub 14419  df-join 14421  df-lat 14463  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-lhyp 30624
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