Metamath Proof Explorer

Theorem cdleme39n

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. E , Y , G , Z serve as f(t), f(u), f_t( R ), f_t( S ). Put hypotheses of cdleme38n in convention of cdleme32sn1awN . TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013)

Ref Expression
Hypotheses cdleme39.l 
|- .<_ = ( le ` K )
cdleme39.j 
|- .\/ = ( join ` K )
cdleme39.m 
|- ./\ = ( meet ` K )
cdleme39.a 
|- A = ( Atoms ` K )
cdleme39.h 
|- H = ( LHyp ` K )
cdleme39.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme39.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme39.g 
|- G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) )
cdleme39.y 
|- Y = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
cdleme39.z 
|- Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( S .\/ u ) ./\ W ) ) )
Assertion cdleme39n 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> G =/= Z )

Proof

Step Hyp Ref Expression
1 cdleme39.l 
 |-  .<_ = ( le ` K )
2 cdleme39.j 
 |-  .\/ = ( join ` K )
3 cdleme39.m 
 |-  ./\ = ( meet ` K )
4 cdleme39.a 
 |-  A = ( Atoms ` K )
5 cdleme39.h 
 |-  H = ( LHyp ` K )
6 cdleme39.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme39.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
8 cdleme39.g 
 |-  G = ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) )
9 cdleme39.y 
 |-  Y = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
10 cdleme39.z 
 |-  Z = ( ( P .\/ Q ) ./\ ( Y .\/ ( ( S .\/ u ) ./\ W ) ) )
11 eqid 
 |-  ( ( t .\/ E ) ./\ W ) = ( ( t .\/ E ) ./\ W )
12 eqid 
 |-  ( ( u .\/ Y ) ./\ W ) = ( ( u .\/ Y ) ./\ W )
13 eqid 
 |-  ( ( R .\/ ( ( t .\/ E ) ./\ W ) ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) = ( ( R .\/ ( ( t .\/ E ) ./\ W ) ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
14 eqid 
 |-  ( ( S .\/ ( ( u .\/ Y ) ./\ W ) ) ./\ ( Y .\/ ( ( u .\/ S ) ./\ W ) ) ) = ( ( S .\/ ( ( u .\/ Y ) ./\ W ) ) ./\ ( Y .\/ ( ( u .\/ S ) ./\ W ) ) )
15 1 2 3 4 5 6 7 9 11 12 13 14 cdleme38n 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( R .\/ ( ( t .\/ E ) ./\ W ) ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) =/= ( ( S .\/ ( ( u .\/ Y ) ./\ W ) ) ./\ ( Y .\/ ( ( u .\/ S ) ./\ W ) ) ) )
16 simp11 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( K e. HL /\ W e. H ) )
17 simp12l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> P e. A )
18 simp13l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> Q e. A )
19 simp22l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R e. A )
20 simp22r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> -. R .<_ W )
21 simp311 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R .<_ ( P .\/ Q ) )
22 simp32l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( t e. A /\ -. t .<_ W ) )
23 1 2 3 4 5 6 7 8 11 cdleme39a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( R e. A /\ -. R .<_ W ) /\ ( R .<_ ( P .\/ Q ) /\ ( t e. A /\ -. t .<_ W ) ) ) -> G = ( ( R .\/ ( ( t .\/ E ) ./\ W ) ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) )
24 16 17 18 19 20 21 22 23 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> G = ( ( R .\/ ( ( t .\/ E ) ./\ W ) ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) )
25 simp23l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> S e. A )
26 simp23r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> -. S .<_ W )
27 simp312 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> S .<_ ( P .\/ Q ) )
28 simp33l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( u e. A /\ -. u .<_ W ) )
29 1 2 3 4 5 6 9 10 12 cdleme39a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ W ) /\ ( S .<_ ( P .\/ Q ) /\ ( u e. A /\ -. u .<_ W ) ) ) -> Z = ( ( S .\/ ( ( u .\/ Y ) ./\ W ) ) ./\ ( Y .\/ ( ( u .\/ S ) ./\ W ) ) ) )
30 16 17 18 25 26 27 28 29 syl322anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> Z = ( ( S .\/ ( ( u .\/ Y ) ./\ W ) ) ./\ ( Y .\/ ( ( u .\/ S ) ./\ W ) ) ) )
31 15 24 30 3netr4d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ R =/= S ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> G =/= Z )