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Theorem cdlemf1 35849
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 1085 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> K e. HL )
2 simp3l 1089 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> P e. A )
3 simp2l 1087 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> U e. A )
4 simp2r 1088 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> U .<_ W )
5 simp3r 1090 . . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. A /\ U .<_ W ) /\ ( P e. A /\ -. P .<_ W ) ) -> -. P .<_ W )
6 nbrne2 4673 . . . . 5 |- ( ( U .<_ W /\ -. P .<_ W ) -> U =/= P )
76necomd 2849 . . . 4 |- ( ( U .<_ W /\ -. P .<_ W ) -> P =/= U )
84, 5, 7syl2anc 693 . . 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 34672 . . 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 1329 . 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 1097 . . . . . 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 2849 . . . . 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 1177 . . . . . 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 1175 . . . . . . . 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 1172 . . . . . . . . . . 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 34650 . . . . . . . . . . 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 2622 . . . . . . . . . . . 12 |- ( Base ` K ) = ( Base ` K )
2221, 11atbase 34576 . . . . . . . . . . 11 |- ( q e. A -> q e. ( Base ` K ) )
23223ad2ant2 1083 . . . . . . . . . 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 1174 . . . . . . . . . . 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 34576 . . . . . . . . . . 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 1173 . . . . . . . . . . 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 35284 . . . . . . . . . . 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 17062 . . . . . . . . . 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 1328 . . . . . . . . 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 219 . . . . . . . 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 710 . . . . . . 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 1099 . . . . . . . . 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 34646 . . . . . . . . . . 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 1062 . . . . . . . . . 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 1176 . . . . . . . . . 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 1098 . . . . . . . . . 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 34624 . . . . . . . . . 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 1339 . . . . . . . . 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 34576 . . . . . . . . . 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 34653 . . . . . . . . . 10 |- ( ( K e. HL /\ q e. A /\ U e. A ) -> ( q .\/ U ) e. ( Base ` K ) )
4718, 38, 24, 46syl3anc 1326 . . . . . . . . 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 17056 . . . . . . . . 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 1328 . . . . . . . 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 711 . . . . . . 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 47 . . . . . 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 189 . . . . 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 34623 . . . . . . 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 1339 . . . . . 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 1242 . . . 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 1264 . . 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 3015 . 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   CvLatclc 34552   HLchlt 34637   LHypclh 35270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274
This theorem is referenced by:  cdlemf2  35850  cdlemg5  35893
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