Metamath Proof Explorer


Theorem cdleme11l

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme11 . (Contributed by NM, 15-Jun-2012)

Ref Expression
Hypotheses cdleme12.l
|- .<_ = ( le ` K )
cdleme12.j
|- .\/ = ( join ` K )
cdleme12.m
|- ./\ = ( meet ` K )
cdleme12.a
|- A = ( Atoms ` K )
cdleme12.h
|- H = ( LHyp ` K )
cdleme12.u
|- U = ( ( P .\/ Q ) ./\ W )
cdleme12.f
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme12.g
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
Assertion cdleme11l
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> F =/= G )

Proof

Step Hyp Ref Expression
1 cdleme12.l
 |-  .<_ = ( le ` K )
2 cdleme12.j
 |-  .\/ = ( join ` K )
3 cdleme12.m
 |-  ./\ = ( meet ` K )
4 cdleme12.a
 |-  A = ( Atoms ` K )
5 cdleme12.h
 |-  H = ( LHyp ` K )
6 cdleme12.u
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme12.f
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme12.g
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 simp11
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( K e. HL /\ W e. H ) )
10 simp12
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11 simp13l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> Q e. A )
12 simp21
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( S e. A /\ -. S .<_ W ) )
13 simp22l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> T e. A )
14 simp23l
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> P =/= Q )
15 simp23r
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> S =/= T )
16 simp31
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. S .<_ ( P .\/ Q ) )
17 simp33
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> U .<_ ( S .\/ T ) )
18 eqid
 |-  ( ( P .\/ S ) ./\ W ) = ( ( P .\/ S ) ./\ W )
19 eqid
 |-  ( ( P .\/ T ) ./\ W ) = ( ( P .\/ T ) ./\ W )
20 1 2 3 4 5 6 18 19 cdleme11e
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ -. S .<_ W ) /\ T e. A /\ P =/= Q ) /\ ( S =/= T /\ -. S .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) ./\ W ) =/= ( ( P .\/ T ) ./\ W ) )
21 9 10 11 12 13 14 15 16 17 20 syl333anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) ./\ W ) =/= ( ( P .\/ T ) ./\ W ) )
22 oveq2
 |-  ( F = G -> ( Q .\/ F ) = ( Q .\/ G ) )
23 22 oveq1d
 |-  ( F = G -> ( ( Q .\/ F ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
24 23 adantl
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ F = G ) -> ( ( Q .\/ F ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
25 simp13
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
26 1 2 3 4 5 6 18 6 7 cdleme11k
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ S ) ./\ W ) = ( ( Q .\/ F ) ./\ W ) )
27 9 10 25 12 14 16 26 syl132anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ S ) ./\ W ) = ( ( Q .\/ F ) ./\ W ) )
28 27 adantr
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ F = G ) -> ( ( P .\/ S ) ./\ W ) = ( ( Q .\/ F ) ./\ W ) )
29 simp22
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( T e. A /\ -. T .<_ W ) )
30 simp32
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> -. T .<_ ( P .\/ Q ) )
31 1 2 3 4 5 6 19 6 8 cdleme11k
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ -. T .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ T ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
32 9 10 25 29 14 30 31 syl132anc
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( P .\/ T ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
33 32 adantr
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ F = G ) -> ( ( P .\/ T ) ./\ W ) = ( ( Q .\/ G ) ./\ W ) )
34 24 28 33 3eqtr4d
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) /\ F = G ) -> ( ( P .\/ S ) ./\ W ) = ( ( P .\/ T ) ./\ W ) )
35 34 ex
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( F = G -> ( ( P .\/ S ) ./\ W ) = ( ( P .\/ T ) ./\ W ) ) )
36 35 necon3d
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> ( ( ( P .\/ S ) ./\ W ) =/= ( ( P .\/ T ) ./\ W ) -> F =/= G ) )
37 21 36 mpd
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ S =/= T ) ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) ) -> F =/= G )