Metamath Proof Explorer

Theorem llnmod1i2

Description: Version of modular law pmod1i that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join P .\/ Q ). (Contributed by NM, 16-Sep-2012) (Revised by Mario Carneiro, 10-May-2013)

Ref Expression
Hypotheses atmod.b 
|- B = ( Base ` K )
atmod.l 
|- .<_ = ( le ` K )
atmod.j 
|- .\/ = ( join ` K )
atmod.m 
|- ./\ = ( meet ` K )
atmod.a 
|- A = ( Atoms ` K )
Assertion llnmod1i2 
|- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) = ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) )

Proof

Step Hyp Ref Expression
1 atmod.b 
 |-  B = ( Base ` K )
2 atmod.l 
 |-  .<_ = ( le ` K )
3 atmod.j 
 |-  .\/ = ( join ` K )
4 atmod.m 
 |-  ./\ = ( meet ` K )
5 atmod.a 
 |-  A = ( Atoms ` K )
6 simpl1 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> K e. HL )
7 simpl2 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> X e. B )
8 simprl 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> P e. A )
9 simprr 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Q e. A )
10 eqid 
 |-  ( pmap ` K ) = ( pmap ` K )
11 eqid 
 |-  ( +P ` K ) = ( +P ` K )
12 1 3 5 10 11 pmapjlln1 
 |-  ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) )
13 6 7 8 9 12 syl13anc 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) )
14 6 hllatd 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> K e. Lat )
15 1 5 atbase 
 |-  ( P e. A -> P e. B )
16 8 15 syl 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> P e. B )
17 1 5 atbase 
 |-  ( Q e. A -> Q e. B )
18 9 17 syl 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Q e. B )
19 1 3 latjcl 
 |-  ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
20 14 16 18 19 syl3anc 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
21 simpl3 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> Y e. B )
22 1 2 3 4 10 11 hlmod1i 
 |-  ( ( K e. HL /\ ( X e. B /\ ( P .\/ Q ) e. B /\ Y e. B ) ) -> ( ( X .<_ Y /\ ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
23 6 7 20 21 22 syl13anc 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( ( X .<_ Y /\ ( ( pmap ` K ) ` ( X .\/ ( P .\/ Q ) ) ) = ( ( ( pmap ` K ) ` X ) ( +P ` K ) ( ( pmap ` K ) ` ( P .\/ Q ) ) ) ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
24 13 23 mpan2d 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) ) -> ( X .<_ Y -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) ) )
25 24 3impia 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) = ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) )
26 25 eqcomd 
 |-  ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ ( P e. A /\ Q e. A ) /\ X .<_ Y ) -> ( X .\/ ( ( P .\/ Q ) ./\ Y ) ) = ( ( X .\/ ( P .\/ Q ) ) ./\ Y ) )