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Metamath Proof Explorer

Theorem cdleme21

Description: Part of proof of Lemma E in Crawley p. 113, 3rd line on p. 115. D , F , N , Y , G , O represent s_2, f(s), f_s(r), t_2, f(t), f_t(r) respectively. Combine cdleme18d and cdleme21j to eliminate existence condition, proving f_s(r) = f_t(r) with fewer conditions. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l 
|- .<_ = ( le ` K )
cdleme21.j 
|- .\/ = ( join ` K )
cdleme21.m 
|- ./\ = ( meet ` K )
cdleme21.a 
|- A = ( Atoms ` K )
cdleme21.h 
|- H = ( LHyp ` K )
cdleme21.u 
|- U = ( ( P .\/ Q ) ./\ W )
cdleme21.f 
|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
cdleme21g.g 
|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
cdleme21g.d 
|- D = ( ( R .\/ S ) ./\ W )
cdleme21g.y 
|- Y = ( ( R .\/ T ) ./\ W )
cdleme21g.n 
|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
cdleme21g.o 
|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion cdleme21 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O )

Proof

Step Hyp Ref Expression
1 cdleme21.l 
 |-  .<_ = ( le ` K )
2 cdleme21.j 
 |-  .\/ = ( join ` K )
3 cdleme21.m 
 |-  ./\ = ( meet ` K )
4 cdleme21.a 
 |-  A = ( Atoms ` K )
5 cdleme21.h 
 |-  H = ( LHyp ` K )
6 cdleme21.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 cdleme21.f 
 |-  F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8 cdleme21g.g 
 |-  G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9 cdleme21g.d 
 |-  D = ( ( R .\/ S ) ./\ W )
10 cdleme21g.y 
 |-  Y = ( ( R .\/ T ) ./\ W )
11 cdleme21g.n 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12 cdleme21g.o 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
14 simpl2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) )
15 simpl3l 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( P =/= Q /\ S =/= T ) )
16 simpl3r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) )
17 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
18 1 2 3 4 5 6 7 8 9 10 11 12 cdleme21j 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> N = O )
19 13 14 15 16 17 18 syl113anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> N = O )
20 simpl1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
21 simpl2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) )
22 simp3ll 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> P =/= Q )
23 22 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> P =/= Q )
24 simp3r3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> R .<_ ( P .\/ Q ) )
25 simp3r1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> -. S .<_ ( P .\/ Q ) )
26 simp3r2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> -. T .<_ ( P .\/ Q ) )
27 24 25 26 3jca 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) )
28 27 adantr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) )
29 simpr 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
30 9 oveq2i 
 |-  ( F .\/ D ) = ( F .\/ ( ( R .\/ S ) ./\ W ) )
31 30 oveq2i 
 |-  ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
32 11 31 eqtri 
 |-  N = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
33 10 oveq2i 
 |-  ( G .\/ Y ) = ( G .\/ ( ( R .\/ T ) ./\ W ) )
34 33 oveq2i 
 |-  ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) = ( ( P .\/ Q ) ./\ ( G .\/ ( ( R .\/ T ) ./\ W ) ) )
35 12 34 eqtri 
 |-  O = ( ( P .\/ Q ) ./\ ( G .\/ ( ( R .\/ T ) ./\ W ) ) )
36 1 2 3 4 5 6 7 32 8 35 cdleme18d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( P =/= Q /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> N = O )
37 20 21 23 28 29 36 syl113anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> N = O )
38 19 37 pm2.61dan 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) /\ R .<_ ( P .\/ Q ) ) ) ) -> N = O )