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Theorem cdlemf1 29654
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 984 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
2 simp3l 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A
)
3 simp2l 986 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  e.  A
)
4 simp2r 987 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  U  .<_  W )
5 simp3r 989 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  A  /\  U  .<_  W )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
6 nbrne2 3938 . . . . 5  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  U  =/=  P
)
76necomd 2495 . . . 4  |-  ( ( U  .<_  W  /\  -.  P  .<_  W )  ->  P  =/=  U
)
84, 5, 7syl2anc 645 . . 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 28479 . . 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 1190 . 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 996 . . . . . 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 2495 . . . . 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 1076 . . . . . 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 1074 . . . . . . . 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 1071 . . . . . . . . . . 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 28457 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
2018, 19syl 17 . . . . . . . . . 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 2253 . . . . . . . . . . . 12  |-  ( Base `  K )  =  (
Base `  K )
2221, 11atbase 28383 . . . . . . . . . . 11  |-  ( q  e.  A  ->  q  e.  ( Base `  K
) )
23223ad2ant2 982 . . . . . . . . . 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 1073 . . . . . . . . . . 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 28383 . . . . . . . . . . 11  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2624, 25syl 17 . . . . . . . . . 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 1072 . . . . . . . . . . 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 29091 . . . . . . . . . . 11  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3027, 29syl 17 . . . . . . . . . 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 14012 . . . . . . . . . 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 1189 . . . . . . . . 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 200 . . . . . . . 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 658 . . . . . . 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 998 . . . . . . . . 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 28453 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  CvLat )
3718, 36syl 17 . . . . . . . . . 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 961 . . . . . . . . . 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 1075 . . . . . . . . . 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 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  =/=  U )
419, 10, 11cvlatexch2 28431 . . . . . . . . . 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 1200 . . . . . . . . 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 16 . . . . . . . 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 28383 . . . . . . . . . 10  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
4539, 44syl 17 . . . . . . . . 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 28460 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  q  e.  A  /\  U  e.  A )  ->  ( q  .\/  U
)  e.  ( Base `  K ) )
4718, 38, 24, 46syl3anc 1187 . . . . . . . . 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 14006 . . . . . . . . 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 1189 . . . . . . . 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 659 . . . . . . 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 28430 . . . . . . 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 1200 . . . . . 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 16 . . . . 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 1137 . . . 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 1155 . . 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 2615 . 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 16 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 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   E.wrex 2510   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   Latclat 13995   Atomscatm 28357   CvLatclc 28359   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdlemf2  29655  cdlemg5  29698
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-join 13954  df-lat 13996  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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