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Metamath Proof Explorer

Theorem 4atexlem7

Description: Whenever there are at least 4 atoms under P .\/ Q (specifically, P , Q , r , and ( P .\/ Q ) ./\ W ), there are also at least 4 atoms under P .\/ S . This proves the statement in Lemma E of Crawley p. 114, last line, "...p \/ q/0 and hence p \/ s/0 contains at least four atoms..." Note that by cvlsupr2 , our ( P .\/ r ) = ( Q .\/ r ) is a shorter way to express r =/= P /\ r =/= Q /\ r .<_ ( P .\/ Q ) . With a longer proof, the condition -. S .<_ ( P .\/ Q ) could be eliminated (see 4atex ), although for some purposes this more restricted lemma may be adequate. (Contributed by NM, 25-Nov-2012)

Ref Expression
Hypotheses 4that.l 
|- .<_ = ( le ` K )
4that.j 
|- .\/ = ( join ` K )
4that.a 
|- A = ( Atoms ` K )
4that.h 
|- H = ( LHyp ` K )
Assertion 4atexlem7 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ 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 simp11l 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( K e. HL /\ W e. H ) )
6 simp1r1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) -> ( P e. A /\ -. P .<_ W ) )
7 6 3ad2ant1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( P e. A /\ -. P .<_ W ) )
8 simp1r2 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) -> ( Q e. A /\ -. Q .<_ W ) )
9 8 3ad2ant1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
10 simp2 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> r e. A )
11 simp3l 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> -. r .<_ W )
12 10 11 jca 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( r e. A /\ -. r .<_ W ) )
13 simp1r3 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) -> S e. A )
14 13 3ad2ant1 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> S e. A )
15 simp3r 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> ( P .\/ r ) = ( Q .\/ r ) )
16 simp12 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> P =/= Q )
17 simp13 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> -. S .<_ ( P .\/ Q ) )
18 eqid 
 |-  ( meet ` K ) = ( meet ` K )
19 1 2 18 3 4 4atexlemex6 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( r e. A /\ -. r .<_ W ) /\ S e. A ) /\ ( ( P .\/ r ) = ( Q .\/ r ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
20 5 7 9 12 14 15 16 17 19 syl323anc 
 |-  ( ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) /\ r e. A /\ ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )
21 20 rexlimdv3a 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) /\ P =/= Q /\ -. S .<_ ( P .\/ Q ) ) -> ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) )
22 21 3exp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) -> ( P =/= Q -> ( -. S .<_ ( P .\/ Q ) -> ( E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) ) )
23 22 3impd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) ) -> ( ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) ) )
24 23 3impia 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ S e. A ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> E. z e. A ( -. z .<_ W /\ ( P .\/ z ) = ( S .\/ z ) ) )