Metamath Proof Explorer

Theorem cdleme39a

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 ) ) )
cdleme39a.v 
|- V = ( ( t .\/ E ) ./\ W )
Assertion 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 .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) )

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 cdleme39a.v 
 |-  V = ( ( t .\/ E ) ./\ W )
10 simp11 
 |-  ( ( ( ( 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 ) ) ) -> ( K e. HL /\ W e. H ) )
11 simp12 
 |-  ( ( ( ( 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 ) ) ) -> P e. A )
12 simp13 
 |-  ( ( ( ( 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 ) ) ) -> Q e. A )
13 simp2 
 |-  ( ( ( ( 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 ) ) ) -> ( R e. A /\ -. R .<_ W ) )
14 simp3l 
 |-  ( ( ( ( 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 ) ) ) -> R .<_ ( P .\/ Q ) )
15 1 2 3 4 5 6 cdleme4 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( R .\/ U ) )
16 10 11 12 13 14 15 syl131anc 
 |-  ( ( ( ( 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 ) ) ) -> ( P .\/ Q ) = ( R .\/ U ) )
17 simp3r 
 |-  ( ( ( ( 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 ) ) ) -> ( t e. A /\ -. t .<_ W ) )
18 1 2 3 4 5 6 7 cdleme2 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( t e. A /\ -. t .<_ W ) ) ) -> ( ( t .\/ E ) ./\ W ) = U )
19 10 11 12 17 18 syl13anc 
 |-  ( ( ( ( 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 ) ) ) -> ( ( t .\/ E ) ./\ W ) = U )
20 9 19 eqtrid 
 |-  ( ( ( ( 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 ) ) ) -> V = U )
21 20 oveq2d 
 |-  ( ( ( ( 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 ) ) ) -> ( R .\/ V ) = ( R .\/ U ) )
22 16 21 eqtr4d 
 |-  ( ( ( ( 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 ) ) ) -> ( P .\/ Q ) = ( R .\/ V ) )
23 simp11l 
 |-  ( ( ( ( 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 ) ) ) -> K e. HL )
24 simp2l 
 |-  ( ( ( ( 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 ) ) ) -> R e. A )
25 simp3rl 
 |-  ( ( ( ( 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 ) ) ) -> t e. A )
26 2 4 hlatjcom 
 |-  ( ( K e. HL /\ R e. A /\ t e. A ) -> ( R .\/ t ) = ( t .\/ R ) )
27 23 24 25 26 syl3anc 
 |-  ( ( ( ( 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 ) ) ) -> ( R .\/ t ) = ( t .\/ R ) )
28 27 oveq1d 
 |-  ( ( ( ( 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 ) ) ) -> ( ( R .\/ t ) ./\ W ) = ( ( t .\/ R ) ./\ W ) )
29 28 oveq2d 
 |-  ( ( ( ( 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 ) ) ) -> ( E .\/ ( ( R .\/ t ) ./\ W ) ) = ( E .\/ ( ( t .\/ R ) ./\ W ) ) )
30 22 29 oveq12d 
 |-  ( ( ( ( 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 ) ) ) -> ( ( P .\/ Q ) ./\ ( E .\/ ( ( R .\/ t ) ./\ W ) ) ) = ( ( R .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) )
31 8 30 eqtrid 
 |-  ( ( ( ( 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 .\/ V ) ./\ ( E .\/ ( ( t .\/ R ) ./\ W ) ) ) )