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Theorem cdlemf1 31372
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 979 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 simp3l 983 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A
)
3 simp2l 981 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  A
)
4 simp2r 982 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  .<_  W )
5 simp3r 984 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
6 nbrne2 4057 . . . . 5  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
76necomd 2542 . . . 4  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  P  =/=  U
)
84, 5, 7syl2anc 642 . . 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 30197 . . 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 1185 . 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 991 . . . . . 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 2542 . . . . 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 1071 . . . . . 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 1069 . . . . . . . 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 1066 . . . . . . . . . . 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 30175 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
2018, 19syl 15 . . . . . . . . . 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 2296 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2221, 11atbase 30101 . . . . . . . . . . 11  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
23223ad2ant2 977 . . . . . . . . . 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 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 ) ) )  ->  U  e.  A )
2521, 11atbase 30101 . . . . . . . . . . 11  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2624, 25syl 15 . . . . . . . . . 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 1067 . . . . . . . . . . 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 30809 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3027, 29syl 15 . . . . . . . . . 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 14184 . . . . . . . . . 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 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  .<_  W  /\  U  .<_  W )  <->  ( q  .\/  U )  .<_  W ) )
3332biimpd 198 . . . . . . . 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 655 . . . . . . 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 993 . . . . . . . . 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 30171 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  CvLat )
3718, 36syl 15 . . . . . . . . . 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 956 . . . . . . . . . 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 1070 . . . . . . . . . 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 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  =/=  U )
419, 10, 11cvlatexch2 30149 . . . . . . . . . 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 1195 . . . . . . . . 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 14 . . . . . . . 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 30101 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4539, 44syl 15 . . . . . . . . 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 30178 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  q  e.  A  /\  U  e.  A )  ->  ( q  .\/  U
)  e.  ( Base `  K ) )
4718, 38, 24, 46syl3anc 1182 . . . . . . . . 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 14178 . . . . . . . . 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 1184 . . . . . . . 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 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  .\/  U
)  .<_  W  ->  P  .<_  W ) )
5134, 50syld 40 . . . . . 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 168 . . . . 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 30148 . . . . . . 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 1195 . . . . . 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 14 . . . . 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 1132 . . . 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 1150 . . 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 2666 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   Latclat 14167   Atomscatm 30075   CvLatclc 30077   HLchlt 30162   LHypclh 30795
This theorem is referenced by:  cdlemf2  31373  cdlemg5  31416
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  df-lhyp 30799
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