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Metamath Proof Explorer

Theorem cvrat42

Description: Commuted version of cvrat4 . (Contributed by NM, 28-Jan-2012)

Ref Expression
Hypotheses cvrat4.b 
|- B = ( Base ` K )
cvrat4.l 
|- .<_ = ( le ` K )
cvrat4.j 
|- .\/ = ( join ` K )
cvrat4.z 
|- .0. = ( 0. ` K )
cvrat4.a 
|- A = ( Atoms ` K )
Assertion cvrat42 
|- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ P .<_ ( X .\/ Q ) ) -> E. r e. A ( r .<_ X /\ P .<_ ( r .\/ Q ) ) ) )

Proof

Step Hyp Ref Expression
1 cvrat4.b 
 |-  B = ( Base ` K )
2 cvrat4.l 
 |-  .<_ = ( le ` K )
3 cvrat4.j 
 |-  .\/ = ( join ` K )
4 cvrat4.z 
 |-  .0. = ( 0. ` K )
5 cvrat4.a 
 |-  A = ( Atoms ` K )
6 1 2 3 4 5 cvrat4 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ P .<_ ( X .\/ Q ) ) -> E. r e. A ( r .<_ X /\ P .<_ ( Q .\/ r ) ) ) )
7 hllat 
 |-  ( K e. HL -> K e. Lat )
8 7 ad2antrr 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> K e. Lat )
9 simplr3 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> Q e. A )
10 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
11 9 10 syl 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> Q e. B )
12 1 5 atbase 
 |-  ( r e. A -> r e. B )
13 12 adantl 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> r e. B )
14 1 3 latjcom 
 |-  ( ( K e. Lat /\ Q e. B /\ r e. B ) -> ( Q .\/ r ) = ( r .\/ Q ) )
15 8 11 13 14 syl3anc 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> ( Q .\/ r ) = ( r .\/ Q ) )
16 15 breq2d 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> ( P .<_ ( Q .\/ r ) <-> P .<_ ( r .\/ Q ) ) )
17 16 anbi2d 
 |-  ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ r e. A ) -> ( ( r .<_ X /\ P .<_ ( Q .\/ r ) ) <-> ( r .<_ X /\ P .<_ ( r .\/ Q ) ) ) )
18 17 rexbidva 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( E. r e. A ( r .<_ X /\ P .<_ ( Q .\/ r ) ) <-> E. r e. A ( r .<_ X /\ P .<_ ( r .\/ Q ) ) ) )
19 6 18 sylibd 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ P .<_ ( X .\/ Q ) ) -> E. r e. A ( r .<_ X /\ P .<_ ( r .\/ Q ) ) ) )