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Metamath Proof Explorer

Theorem dihjatc2N

Description: Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihjatc1.b 
|- B = ( Base ` K )
dihjatc1.l 
|- .<_ = ( le ` K )
dihjatc1.h 
|- H = ( LHyp ` K )
dihjatc1.j 
|- .\/ = ( join ` K )
dihjatc1.m 
|- ./\ = ( meet ` K )
dihjatc1.a 
|- A = ( Atoms ` K )
dihjatc1.u 
|- U = ( ( DVecH ` K ) ` W )
dihjatc1.s 
|- .(+) = ( LSSum ` U )
dihjatc1.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihjatc2N 
|- ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( Q .\/ ( X ./\ Y ) ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) )

Proof

Step Hyp Ref Expression
1 dihjatc1.b 
 |-  B = ( Base ` K )
2 dihjatc1.l 
 |-  .<_ = ( le ` K )
3 dihjatc1.h 
 |-  H = ( LHyp ` K )
4 dihjatc1.j 
 |-  .\/ = ( join ` K )
5 dihjatc1.m 
 |-  ./\ = ( meet ` K )
6 dihjatc1.a 
 |-  A = ( Atoms ` K )
7 dihjatc1.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihjatc1.s 
 |-  .(+) = ( LSSum ` U )
9 dihjatc1.i 
 |-  I = ( ( DIsoH ` K ) ` W )
10 simp11l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> K e. HL )
11 10 hllatd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> K e. Lat )
12 simp2l 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Q e. A )
13 1 6 atbase 
 |-  ( Q e. A -> Q e. B )
14 12 13 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Q e. B )
15 simp12 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> X e. B )
16 simp13 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> Y e. B )
17 1 5 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
18 11 15 16 17 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( X ./\ Y ) e. B )
19 1 4 latjcom 
 |-  ( ( K e. Lat /\ Q e. B /\ ( X ./\ Y ) e. B ) -> ( Q .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ Q ) )
20 11 14 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( Q .\/ ( X ./\ Y ) ) = ( ( X ./\ Y ) .\/ Q ) )
21 20 fveq2d 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( Q .\/ ( X ./\ Y ) ) ) = ( I ` ( ( X ./\ Y ) .\/ Q ) ) )
22 1 2 3 4 5 6 7 8 9 dihjatc1 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( ( X ./\ Y ) .\/ Q ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) )
23 21 22 eqtrd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` ( Q .\/ ( X ./\ Y ) ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) )