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Theorem cdlemf1 36409
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 1020 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 simp3l 1024 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A
)
3 simp2l 1022 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  A
)
4 simp2r 1023 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  .<_  W )
5 simp3r 1025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
6 nbrne2 4474 . . . . 5  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
76necomd 2728 . . . 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 35232 . . 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 1231 . 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 1032 . . . . . 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 2728 . . . . 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 1112 . . . . . 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 1110 . . . . . . . 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 1107 . . . . . . . . . . 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 35210 . . . . . . . . . . 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 2457 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2221, 11atbase 35136 . . . . . . . . . . 11  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
23223ad2ant2 1018 . . . . . . . . . 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 1109 . . . . . . . . . . 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 35136 . . . . . . . . . . 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 1108 . . . . . . . . . . 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 35844 . . . . . . . . . . 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 15819 . . . . . . . . . 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 1230 . . . . . . . . 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 1034 . . . . . . . . 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 35206 . . . . . . . . . . 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 997 . . . . . . . . . 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 1111 . . . . . . . . . 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 1033 . . . . . . . . . 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 35184 . . . . . . . . . 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 1241 . . . . . . . . 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 35136 . . . . . . . . . 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 35213 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  q  e.  A  /\  U  e.  A )  ->  ( q  .\/  U
)  e.  ( Base `  K ) )
4718, 38, 24, 46syl3anc 1228 . . . . . . . . 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 15813 . . . . . . . . 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 1230 . . . . . . . 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 35183 . . . . . . 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 1241 . . . . . 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 1176 . . . 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 1195 . . 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 2929 . 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   E.wrex 2808   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   lecple 14719   joincjn 15700   Latclat 15802   Atomscatm 35110   CvLatclc 35112   HLchlt 35197   LHypclh 35830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-preset 15684  df-poset 15702  df-plt 15715  df-lub 15731  df-glb 15732  df-join 15733  df-meet 15734  df-p0 15796  df-lat 15803  df-covers 35113  df-ats 35114  df-atl 35145  df-cvlat 35169  df-hlat 35198  df-lhyp 35834
This theorem is referenced by:  cdlemf2  36410  cdlemg5  36453
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