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Metamath Proof Explorer

Theorem cdleme38m

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. (Contributed by NM, 13-Mar-2013)

Ref Expression
Hypotheses cdleme38.l 
|- .<_ = ( le ` K )
cdleme38.j 
|- .\/ = ( join ` K )
cdleme38.m 
|- ./\ = ( meet ` K )
cdleme38.a 
|- A = ( Atoms ` K )
cdleme38.h 
|- H = ( LHyp ` K )
cdleme38.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme38.e 
|- E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
cdleme38.d 
|- D = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
cdleme38.v 
|- V = ( ( t .\/ E ) ./\ W )
cdleme38.x 
|- X = ( ( u .\/ D ) ./\ W )
cdleme38.f 
|- F = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
cdleme38.g 
|- G = ( ( S .\/ X ) ./\ ( D .\/ ( ( u .\/ S ) ./\ W ) ) )
Assertion cdleme38m 
|- ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R = S )

Proof

Step Hyp Ref Expression
1 cdleme38.l 
 |-  .<_ = ( le ` K )
2 cdleme38.j 
 |-  .\/ = ( join ` K )
3 cdleme38.m 
 |-  ./\ = ( meet ` K )
4 cdleme38.a 
 |-  A = ( Atoms ` K )
5 cdleme38.h 
 |-  H = ( LHyp ` K )
6 cdleme38.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme38.e 
 |-  E = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
8 cdleme38.d 
 |-  D = ( ( u .\/ U ) ./\ ( Q .\/ ( ( P .\/ u ) ./\ W ) ) )
9 cdleme38.v 
 |-  V = ( ( t .\/ E ) ./\ W )
10 cdleme38.x 
 |-  X = ( ( u .\/ D ) ./\ W )
11 cdleme38.f 
 |-  F = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
12 cdleme38.g 
 |-  G = ( ( S .\/ X ) ./\ ( D .\/ ( ( u .\/ S ) ./\ W ) ) )
13 simp1 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
14 simp2 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) )
15 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R .<_ ( P .\/ Q ) )
16 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> S .<_ ( P .\/ Q ) )
17 simp313 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F = G )
18 15 16 jca 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) ) )
19 simp32 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) )
20 simp33 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) )
21 eqid 
 |-  ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) )
22 1 2 3 4 5 6 7 8 9 10 21 12 cdleme37m 
 |-  ( ( ( ( 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 ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = G )
23 13 14 18 19 20 22 syl113anc 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) = G )
24 17 23 eqtr4d 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> F = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) )
25 15 16 24 3jca 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ S .<_ ( P .\/ Q ) /\ F = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) ) )
26 eqid 
 |-  ( Base ` K ) = ( Base ` K )
27 26 1 2 3 4 5 6 7 9 11 21 cdleme36m 
 |-  ( ( ( ( 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 ) /\ F = ( ( S .\/ V ) ./\ ( E .\/ ( ( t .\/ S ) ./\ W ) ) ) ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) ) ) -> R = S )
28 13 14 25 19 27 syl112anc 
 |-  ( ( ( ( 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 ) /\ F = G ) /\ ( ( t e. A /\ -. t .<_ W ) /\ -. t .<_ ( P .\/ Q ) ) /\ ( ( u e. A /\ -. u .<_ W ) /\ -. u .<_ ( P .\/ Q ) ) ) ) -> R = S )