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Metamath Proof Explorer

Theorem cdleme21at

Description: Part of proof of Lemma E in Crawley p. 115. (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 )
Assertion cdleme21at 
|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> T =/= z )

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 1 2 3 4 5 6 cdleme21c 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. U .<_ ( S .\/ z ) )
8 7 3adant2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. U .<_ ( S .\/ z ) )
9 simp2r 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> U .<_ ( S .\/ T ) )
10 oveq2 
 |-  ( T = z -> ( S .\/ T ) = ( S .\/ z ) )
11 10 breq2d 
 |-  ( T = z -> ( U .<_ ( S .\/ T ) <-> U .<_ ( S .\/ z ) ) )
12 9 11 syl5ibcom 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( T = z -> U .<_ ( S .\/ z ) ) )
13 12 necon3bd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( -. U .<_ ( S .\/ z ) -> T =/= z ) )
14 8 13 mpd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ Q e. A ) /\ ( ( S e. A /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ U .<_ ( S .\/ T ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> T =/= z )