Metamath Proof Explorer

Theorem 4atexlemex4

Description: Lemma for 4atexlem7 . Show that when C = S , D satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012)

Ref Expression
Hypotheses 4thatlem.ph 
|- ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
4thatlem0.l 
|- .<_ = ( le ` K )
4thatlem0.j 
|- .\/ = ( join ` K )
4thatlem0.m 
|- ./\ = ( meet ` K )
4thatlem0.a 
|- A = ( Atoms ` K )
4thatlem0.h 
|- H = ( LHyp ` K )
4thatlem0.u 
|- U = ( ( P .\/ Q ) ./\ W )
4thatlem0.v 
|- V = ( ( P .\/ S ) ./\ W )
4thatlem0.c 
|- C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
4thatlem0.d 
|- D = ( ( R .\/ T ) ./\ ( P .\/ S ) )
Assertion 4atexlemex4 
|- ( ( ph /\ C = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )

Proof

Step Hyp Ref Expression
1 4thatlem.ph 
 |-  ( ph <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( S e. A /\ ( R e. A /\ -. R .<_ W /\ ( P .\/ R ) = ( Q .\/ R ) ) /\ ( T e. A /\ ( U .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) )
2 4thatlem0.l 
 |-  .<_ = ( le ` K )
3 4thatlem0.j 
 |-  .\/ = ( join ` K )
4 4thatlem0.m 
 |-  ./\ = ( meet ` K )
5 4thatlem0.a 
 |-  A = ( Atoms ` K )
6 4thatlem0.h 
 |-  H = ( LHyp ` K )
7 4thatlem0.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
8 4thatlem0.v 
 |-  V = ( ( P .\/ S ) ./\ W )
9 4thatlem0.c 
 |-  C = ( ( Q .\/ T ) ./\ ( P .\/ S ) )
10 4thatlem0.d 
 |-  D = ( ( R .\/ T ) ./\ ( P .\/ S ) )
11 1 2 3 5 7 4atexlemswapqr 
 |-  ( ph -> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) )
12 1 2 3 4 5 6 7 8 9 10 4atexlemcnd 
 |-  ( ph -> C =/= D )
13 pm13.18 
 |-  ( ( C = S /\ C =/= D ) -> S =/= D )
14 13 necomd 
 |-  ( ( C = S /\ C =/= D ) -> D =/= S )
15 14 expcom 
 |-  ( C =/= D -> ( C = S -> D =/= S ) )
16 12 15 syl 
 |-  ( ph -> ( C = S -> D =/= S ) )
17 biid 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) <-> ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) )
18 eqid 
 |-  ( ( P .\/ R ) ./\ W ) = ( ( P .\/ R ) ./\ W )
19 17 2 3 4 5 6 18 8 10 4atexlemex2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( R e. A /\ -. R .<_ W ) ) /\ ( S e. A /\ ( Q e. A /\ -. Q .<_ W /\ ( P .\/ Q ) = ( R .\/ Q ) ) /\ ( T e. A /\ ( ( ( P .\/ R ) ./\ W ) .\/ T ) = ( V .\/ T ) ) ) /\ ( P =/= R /\ -. S .<_ ( P .\/ R ) ) ) /\ D =/= S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
20 11 16 19 syl6an 
 |-  ( ph -> ( C = S -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) )
21 20 imp 
 |-  ( ( ph /\ C = S ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )