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Metamath Proof Explorer

Theorem 4atex2-0aOLDN

Description: Same as 4atex2 except that S is zero. (Contributed by NM, 27-May-2013) (New usage is discouraged.)

Ref Expression
Hypotheses 4that.l 
|- .<_ = ( le ` K )
4that.j 
|- .\/ = ( join ` K )
4that.a 
|- A = ( Atoms ` K )
4that.h 
|- H = ( LHyp ` K )
Assertion 4atex2-0aOLDN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) )

Proof

Step Hyp Ref Expression
1 4that.l 
 |-  .<_ = ( le ` K )
2 4that.j 
 |-  .\/ = ( join ` K )
3 4that.a 
 |-  A = ( Atoms ` K )
4 4that.h 
 |-  H = ( LHyp ` K )
5 simp32l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> T e. A )
6 simp32r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. T .<_ W )
7 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. HL )
8 hlol 
 |-  ( K e. HL -> K e. OL )
9 7 8 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> K e. OL )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 3 atbase 
 |-  ( T e. A -> T e. ( Base ` K ) )
12 5 11 syl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> T e. ( Base ` K ) )
13 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
14 10 2 13 olj02 
 |-  ( ( K e. OL /\ T e. ( Base ` K ) ) -> ( ( 0. ` K ) .\/ T ) = T )
15 9 12 14 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( 0. ` K ) .\/ T ) = T )
16 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> S = ( 0. ` K ) )
17 16 oveq1d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( S .\/ T ) = ( ( 0. ` K ) .\/ T ) )
18 2 3 hlatjidm 
 |-  ( ( K e. HL /\ T e. A ) -> ( T .\/ T ) = T )
19 7 5 18 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( T .\/ T ) = T )
20 15 17 19 3eqtr4d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( S .\/ T ) = ( T .\/ T ) )
21 breq1 
 |-  ( z = T -> ( z .<_ W <-> T .<_ W ) )
22 21 notbid 
 |-  ( z = T -> ( -. z .<_ W <-> -. T .<_ W ) )
23 oveq2 
 |-  ( z = T -> ( S .\/ z ) = ( S .\/ T ) )
24 oveq2 
 |-  ( z = T -> ( T .\/ z ) = ( T .\/ T ) )
25 23 24 eqeq12d 
 |-  ( z = T -> ( ( S .\/ z ) = ( T .\/ z ) <-> ( S .\/ T ) = ( T .\/ T ) ) )
26 22 25 anbi12d 
 |-  ( z = T -> ( ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) <-> ( -. T .<_ W /\ ( S .\/ T ) = ( T .\/ T ) ) ) )
27 26 rspcev 
 |-  ( ( T e. A /\ ( -. T .<_ W /\ ( S .\/ T ) = ( T .\/ T ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) )
28 5 6 20 27 syl12anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S = ( 0. ` K ) ) /\ ( P =/= Q /\ ( T e. A /\ -. T .<_ W ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( S .\/ z ) = ( T .\/ z ) ) )