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Metamath Proof Explorer

Theorem pmapjlln1

Description: The projective map of the join of a lattice element and a lattice line (expressed as the join Q .\/ R of two atoms). (Contributed by NM, 16-Sep-2012)

Ref Expression
Hypotheses pmapjat.b 
|- B = ( Base ` K )
pmapjat.j 
|- .\/ = ( join ` K )
pmapjat.a 
|- A = ( Atoms ` K )
pmapjat.m 
|- M = ( pmap ` K )
pmapjat.p 
|- .+ = ( +P ` K )
Assertion pmapjlln1 
|- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) )

Proof

Step Hyp Ref Expression
1 pmapjat.b 
 |-  B = ( Base ` K )
2 pmapjat.j 
 |-  .\/ = ( join ` K )
3 pmapjat.a 
 |-  A = ( Atoms ` K )
4 pmapjat.m 
 |-  M = ( pmap ` K )
5 pmapjat.p 
 |-  .+ = ( +P ` K )
6 simpl 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> K e. HL )
7 1 3 4 pmapssat 
 |-  ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ A )
8 7 3ad2antr1 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` X ) C_ A )
9 simpr2 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> Q e. A )
10 1 3 atbase 
 |-  ( Q e. A -> Q e. B )
11 9 10 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> Q e. B )
12 1 3 4 pmapssat 
 |-  ( ( K e. HL /\ Q e. B ) -> ( M ` Q ) C_ A )
13 11 12 syldan 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` Q ) C_ A )
14 simpr3 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> R e. A )
15 1 3 atbase 
 |-  ( R e. A -> R e. B )
16 14 15 syl 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> R e. B )
17 1 3 4 pmapssat 
 |-  ( ( K e. HL /\ R e. B ) -> ( M ` R ) C_ A )
18 16 17 syldan 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` R ) C_ A )
19 3 5 paddass 
 |-  ( ( K e. HL /\ ( ( M ` X ) C_ A /\ ( M ` Q ) C_ A /\ ( M ` R ) C_ A ) ) -> ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
20 6 8 13 18 19 syl13anc 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
21 hllat 
 |-  ( K e. HL -> K e. Lat )
22 21 adantr 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> K e. Lat )
23 simpr1 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> X e. B )
24 1 2 latjcl 
 |-  ( ( K e. Lat /\ X e. B /\ Q e. B ) -> ( X .\/ Q ) e. B )
25 22 23 11 24 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( X .\/ Q ) e. B )
26 1 2 3 4 5 pmapjat1 
 |-  ( ( K e. HL /\ ( X .\/ Q ) e. B /\ R e. A ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) )
27 6 25 14 26 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) )
28 1 2 latjass 
 |-  ( ( K e. Lat /\ ( X e. B /\ Q e. B /\ R e. B ) ) -> ( ( X .\/ Q ) .\/ R ) = ( X .\/ ( Q .\/ R ) ) )
29 22 23 11 16 28 syl13anc 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( X .\/ Q ) .\/ R ) = ( X .\/ ( Q .\/ R ) ) )
30 29 fveq2d 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( ( X .\/ Q ) .\/ R ) ) = ( M ` ( X .\/ ( Q .\/ R ) ) ) )
31 1 2 3 4 5 pmapjat1 
 |-  ( ( K e. HL /\ X e. B /\ Q e. A ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) )
32 31 3adant3r3 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ Q ) ) = ( ( M ` X ) .+ ( M ` Q ) ) )
33 32 oveq1d 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( M ` ( X .\/ Q ) ) .+ ( M ` R ) ) = ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) )
34 27 30 33 3eqtr3d 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( ( M ` X ) .+ ( M ` Q ) ) .+ ( M ` R ) ) )
35 1 2 3 4 5 pmapjat1 
 |-  ( ( K e. HL /\ Q e. B /\ R e. A ) -> ( M ` ( Q .\/ R ) ) = ( ( M ` Q ) .+ ( M ` R ) ) )
36 6 11 14 35 syl3anc 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( Q .\/ R ) ) = ( ( M ` Q ) .+ ( M ` R ) ) )
37 36 oveq2d 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( ( M ` Q ) .+ ( M ` R ) ) ) )
38 20 34 37 3eqtr4d 
 |-  ( ( K e. HL /\ ( X e. B /\ Q e. A /\ R e. A ) ) -> ( M ` ( X .\/ ( Q .\/ R ) ) ) = ( ( M ` X ) .+ ( M ` ( Q .\/ R ) ) ) )