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Metamath Proof Explorer

Theorem cdleme0ex2N

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.)

Ref Expression
Hypotheses 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 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 ) )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 cdleme0.h 
 |-  H = ( LHyp ` K )
6 cdleme0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( K e. HL /\ W e. H ) )
8 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> ( P e. A /\ -. P .<_ W ) )
9 simp2rl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> Q e. A )
10 simp3 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P =/= Q )
11 1 2 3 4 5 6 cdleme0ex1N 
 |-  ( ( ( 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 ) )
12 7 8 9 10 11 syl121anc 
 |-  ( ( ( 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 
 |-  ( ( ( ( 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 
 |-  ( K e. HL -> K e. CvLat )
15 13 14 syl 
 |-  ( ( ( ( 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 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> P e. A )
17 16 3ad2ant1 
 |-  ( ( ( ( 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 )
18 9 3ad2ant1 
 |-  ( ( ( ( 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 
 |-  ( ( ( ( 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 
 |-  ( ( ( ( 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 )
21 4 1 2 cvlsupr2 
 |-  ( ( 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 ) ) ) )
22 15 17 18 19 20 21 syl131anc 
 |-  ( ( ( ( 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 df-3an 
 |-  ( ( u =/= P /\ u =/= Q /\ u .<_ ( P .\/ Q ) ) <-> ( ( u =/= P /\ u =/= Q ) /\ u .<_ ( P .\/ Q ) ) )
24 simp3 
 |-  ( ( ( ( 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 )
25 simp2lr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. P .<_ W )
26 25 3ad2ant1 
 |-  ( ( ( ( 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 )
27 nbrne2 
 |-  ( ( u .<_ W /\ -. P .<_ W ) -> u =/= P )
28 24 26 27 syl2anc 
 |-  ( ( ( ( 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 )
29 simp2rr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ P =/= Q ) -> -. Q .<_ W )
30 29 3ad2ant1 
 |-  ( ( ( ( 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 )
31 nbrne2 
 |-  ( ( u .<_ W /\ -. Q .<_ W ) -> u =/= Q )
32 24 30 31 syl2anc 
 |-  ( ( ( ( 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 )
33 28 32 jca 
 |-  ( ( ( ( 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 ) )
34 33 biantrurd 
 |-  ( ( ( ( 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 ) ) ) )
35 23 34 bitr4id 
 |-  ( ( ( ( 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 ) ) )
36 22 35 bitrd 
 |-  ( ( ( ( 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 ) ) )
37 36 3expia 
 |-  ( ( ( ( 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 ) ) ) )
38 37 pm5.32rd 
 |-  ( ( ( ( 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 ) ) )
39 38 rexbidva 
 |-  ( ( ( 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 ) ) )
40 12 39 mpbird 
 |-  ( ( ( 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 ) )