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Theorem cdleme0ex2N 35511
Description: Part of proof of Lemma E in [Crawley] p. 113. Note that ( P .\/ u ) = ( Q .\/ u ) is a shorter way to express u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdleme0.l |- .<_ = ( le ` K )
cdleme0.j |- .\/ = ( join ` K )
cdleme0.m |- ./\ = ( meet ` K )
cdleme0.a |- A = ( Atoms ` K )
cdleme0.h |- H = ( LHyp ` K )
cdleme0.u |- U = ( ( P .\/ Q ) ./\ W )
Assertion
Ref Expression
cdleme0ex2N |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) )
Distinct variable groups:   u, A   u, .\/   u, .<_   u, P   u, Q   u, U   u, W   u, H   u, K
Allowed substitution hint:   ./\ ( u)
Proof of Theorem cdleme0ex2N
StepHypRef Expression
1 simp1 1061 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
2 simp2l 1087 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
3 simp2rl 1130 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
4 simp3 1063 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
5 cdleme0.l . . . 4 |- .<_ = ( le ` K )
6 cdleme0.j . . . 4 |- .\/ = ( join ` K )
7 cdleme0.m . . . 4 |- ./\ = ( meet ` K )
8 cdleme0.a . . . 4 |- A = ( Atoms ` K )
9 cdleme0.h . . . 4 |- H = ( LHyp ` K )
10 cdleme0.u . . . 4 |- U = ( ( P .\/ Q ) ./\ W )
115, 6, 7, 8, 9, 10cdleme0ex1N 35510 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
121, 2, 3, 4, 11syl121anc 1331 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) )
13 simp11l 1172 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> K e. HL )
14 hlcvl 34646 . . . . . . . 8 |- ( K e. HL -> K e. CvLat )
1513, 14syl 17 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> K e. CvLat )
16 simp2ll 1128 . . . . . . . 8 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
17163ad2ant1 1082 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> P e. A )
1833ad2ant1 1082 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> Q e. A )
19 simp2 1062 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u e. A )
20 simp13 1093 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> P =/= Q )
218, 5, 6cvlsupr2 34630 . . . . . . 7 |- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ u e. A ) /\ P =/= Q ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) ) )
2215, 17, 18, 19, 20, 21syl131anc 1339 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) ) )
23 simp3 1063 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u .<_ W )
24 simp2lr 1129 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
25243ad2ant1 1082 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> -. P .<_ W )
26 nbrne2 4673 . . . . . . . . . 10 |- ( ( u .<_ W /\ -. P .<_ W ) -> u =/= P )
2723, 25, 26syl2anc 693 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u =/= P )
28 simp2rr 1131 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
29283ad2ant1 1082 . . . . . . . . . 10 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> -. Q .<_ W )
30 nbrne2 4673 . . . . . . . . . 10 |- ( ( u .<_ W /\ -. Q .<_ W ) -> u =/= Q )
3123, 29, 30syl2anc 693 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> u =/= Q )
3227, 31jca 554 . . . . . . . 8 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( u =/= P /\ u =/= Q ) )
3332biantrurd 529 . . . . . . 7 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( u .<_ ( P .\/ Q ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) ) )
34 df-3an 1039 . . . . . . 7 |- ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) )
3533, 34syl6rbbr 279 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> u .<_ ( P .\/ Q ) ) )
3622, 35bitrd 268 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A /\ u .<_ W ) -> ( ( P .\/ u ) = ( Q .\/ u ) <-> u .<_ ( P .\/ Q ) ) )
37363expia 1267 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A ) -> ( u .<_ W -> ( ( P .\/ u ) = ( Q .\/ u ) <-> u .<_ ( P .\/ Q ) ) ) )
3837pm5.32rd 672 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) /\ u e. A ) -> ( ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) <-> ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) )
3938rexbidva 3049 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) <-> E. u e. A ( u .<_ ( P .\/ Q ) /\ u .<_ W ) ) )
4012, 39mpbird 247 1 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> E. u e. A ( ( P .\/ u ) = ( Q .\/ u ) /\ u .<_ W ) )
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   lecple 15948   joincjn 16944   meetcmee 16945   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-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274
This theorem is referenced by: (None)
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