This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem cdleme22a

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 3rd line on p. 115. Show that t \/ v = p \/ q implies v = u. (Contributed by NM, 30-Nov-2012)

Ref Expression
Hypotheses cdleme22.l 
|- .<_ = ( le ` K )
cdleme22.j 
|- .\/ = ( join ` K )
cdleme22.m 
|- ./\ = ( meet ` K )
cdleme22.a 
|- A = ( Atoms ` K )
cdleme22.h 
|- H = ( LHyp ` K )
cdleme22.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdleme22a 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V = U )

Proof

Step Hyp Ref Expression
1 cdleme22.l 
 |-  .<_ = ( le ` K )
2 cdleme22.j 
 |-  .\/ = ( join ` K )
3 cdleme22.m 
 |-  ./\ = ( meet ` K )
4 cdleme22.a 
 |-  A = ( Atoms ` K )
5 cdleme22.h 
 |-  H = ( LHyp ` K )
6 cdleme22.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
7 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
8 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
9 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> Q e. A )
10 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> P =/= Q )
11 simp31l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V e. A )
12 simp31r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V .<_ W )
13 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> K e. HL )
14 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> T e. A )
15 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ T e. A /\ V e. A ) -> V .<_ ( T .\/ V ) )
16 13 14 11 15 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V .<_ ( T .\/ V ) )
17 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> ( T .\/ V ) = ( P .\/ Q ) )
18 16 17 breqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V .<_ ( P .\/ Q ) )
19 1 2 3 4 5 6 cdleme22aa 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ P =/= Q ) /\ ( V e. A /\ V .<_ W /\ V .<_ ( P .\/ Q ) ) ) -> V = U )
20 7 8 9 10 11 12 18 19 syl133anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ T e. A ) /\ ( ( V e. A /\ V .<_ W ) /\ P =/= Q /\ ( T .\/ V ) = ( P .\/ Q ) ) ) -> V = U )