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Metamath Proof Explorer

Theorem dihjatc3

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

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 dihjatc3 
|- ( ( ( ( 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 ` ( X ./\ Y ) ) .(+) ( I ` Q ) ) )

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 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 ) ) ) )
11 simp11 
 |-  ( ( ( ( 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 /\ W e. H ) )
12 3 7 11 dvhlmod 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> U e. LMod )
13 lmodabl 
 |-  ( U e. LMod -> U e. Abel )
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 ) ) -> U e. Abel )
15 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
16 15 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
17 12 16 syl 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
18 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 )
19 18 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 )
20 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 )
21 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 )
22 1 5 latmcl 
 |-  ( ( K e. Lat /\ X e. B /\ Y e. B ) -> ( X ./\ Y ) e. B )
23 19 20 21 22 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 )
24 1 3 9 7 15 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X ./\ Y ) e. B ) -> ( I ` ( X ./\ Y ) ) e. ( LSubSp ` U ) )
25 11 23 24 syl2anc 
 |-  ( ( ( ( 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 ) ) e. ( LSubSp ` U ) )
26 17 25 sseldd 
 |-  ( ( ( ( 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 ) ) e. ( SubGrp ` U ) )
27 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 )
28 1 6 atbase 
 |-  ( Q e. A -> Q e. B )
29 27 28 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 )
30 1 3 9 7 15 dihlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ Q e. B ) -> ( I ` Q ) e. ( LSubSp ` U ) )
31 11 29 30 syl2anc 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` Q ) e. ( LSubSp ` U ) )
32 17 31 sseldd 
 |-  ( ( ( ( K e. HL /\ W e. H ) /\ X e. B /\ Y e. B ) /\ ( Q e. A /\ -. Q .<_ W ) /\ ( Q .<_ X /\ ( X ./\ Y ) .<_ W ) ) -> ( I ` Q ) e. ( SubGrp ` U ) )
33 8 lsmcom 
 |-  ( ( U e. Abel /\ ( I ` ( X ./\ Y ) ) e. ( SubGrp ` U ) /\ ( I ` Q ) e. ( SubGrp ` U ) ) -> ( ( I ` ( X ./\ Y ) ) .(+) ( I ` Q ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) )
34 14 26 32 33 syl3anc 
 |-  ( ( ( ( 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 ) ) .(+) ( I ` Q ) ) = ( ( I ` Q ) .(+) ( I ` ( X ./\ Y ) ) ) )
35 10 34 eqtr4d 
 |-  ( ( ( ( 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 ` ( X ./\ Y ) ) .(+) ( I ` Q ) ) )