# Metamath Proof Explorer

## Theorem omlfh3N

Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N . (Contributed by NM, 8-Nov-2011) (New usage is discouraged.)

Ref Expression
Hypotheses omlfh1.b
`|- B = ( Base ` K )`
omlfh1.j
`|- .\/ = ( join ` K )`
omlfh1.m
`|- ./\ = ( meet ` K )`
omlfh1.c
`|- C = ( cm ` K )`
Assertion omlfh3N
`|- ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Y /\ X C Z ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) )`

### Proof

Step Hyp Ref Expression
1 omlfh1.b
` |-  B = ( Base ` K )`
2 omlfh1.j
` |-  .\/ = ( join ` K )`
3 omlfh1.m
` |-  ./\ = ( meet ` K )`
4 omlfh1.c
` |-  C = ( cm ` K )`
5 eqid
` |-  ( oc ` K ) = ( oc ` K )`
6 1 5 4 cmt4N
` |-  ( ( K e. OML /\ X e. B /\ Y e. B ) -> ( X C Y <-> ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) ) )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Y <-> ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) ) )`
8 1 5 4 cmt4N
` |-  ( ( K e. OML /\ X e. B /\ Z e. B ) -> ( X C Z <-> ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X C Z <-> ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) )`
10 7 9 anbi12d
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ X C Z ) <-> ( ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) ) )`
11 simpl
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OML )`
12 omlop
` |-  ( K e. OML -> K e. OP )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OP )`
14 simpr1
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> X e. B )`
15 1 5 opoccl
` |-  ( ( K e. OP /\ X e. B ) -> ( ( oc ` K ) ` X ) e. B )`
16 13 14 15 syl2anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` X ) e. B )`
17 simpr2
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Y e. B )`
18 1 5 opoccl
` |-  ( ( K e. OP /\ Y e. B ) -> ( ( oc ` K ) ` Y ) e. B )`
19 13 17 18 syl2anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` Y ) e. B )`
20 simpr3
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> Z e. B )`
21 1 5 opoccl
` |-  ( ( K e. OP /\ Z e. B ) -> ( ( oc ` K ) ` Z ) e. B )`
22 13 20 21 syl2anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` Z ) e. B )`
23 16 19 22 3jca
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) )`
24 1 2 3 4 omlfh1N
` |-  ( ( K e. OML /\ ( ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) /\ ( ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) ) -> ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) = ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) )`
25 24 fveq2d
` |-  ( ( K e. OML /\ ( ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) /\ ( ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
26 25 3exp
` |-  ( K e. OML -> ( ( ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) -> ( ( ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) ) ) )`
27 11 23 26 sylc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Y ) /\ ( ( oc ` K ) ` X ) C ( ( oc ` K ) ` Z ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) ) )`
28 10 27 sylbid
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X C Y /\ X C Z ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) ) )`
29 28 3impia
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Y /\ X C Z ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
30 omlol
` |-  ( K e. OML -> K e. OL )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OL )`
32 omllat
` |-  ( K e. OML -> K e. Lat )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. Lat )`
34 1 2 latjcl
` |-  ( ( K e. Lat /\ ( ( oc ` K ) ` Y ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) -> ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) e. B )`
35 33 19 22 34 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) e. B )`
36 1 2 3 5 oldmm2
` |-  ( ( K e. OL /\ X e. B /\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( X .\/ ( ( oc ` K ) ` ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) )`
37 31 14 35 36 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( X .\/ ( ( oc ` K ) ` ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) )`
38 1 2 3 5 oldmj4
` |-  ( ( K e. OL /\ Y e. B /\ Z e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) = ( Y ./\ Z ) )`
39 31 17 20 38 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) = ( Y ./\ Z ) )`
40 39 oveq2d
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( ( oc ` K ) ` ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) = ( X .\/ ( Y ./\ Z ) ) )`
41 37 40 eqtr2d
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Y /\ X C Z ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( ( oc ` K ) ` Y ) .\/ ( ( oc ` K ) ` Z ) ) ) ) )`
43 1 3 latmcl
` |-  ( ( K e. Lat /\ ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Y ) e. B ) -> ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) e. B )`
44 33 16 19 43 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) e. B )`
45 1 3 latmcl
` |-  ( ( K e. Lat /\ ( ( oc ` K ) ` X ) e. B /\ ( ( oc ` K ) ` Z ) e. B ) -> ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) e. B )`
46 33 16 22 45 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) e. B )`
47 1 2 3 5 oldmj1
` |-  ( ( K e. OL /\ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) e. B /\ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) e. B ) -> ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) = ( ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) ) ./\ ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
48 31 44 46 47 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) = ( ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) ) ./\ ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
49 1 2 3 5 oldmm4
` |-  ( ( K e. OL /\ X e. B /\ Y e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) ) = ( X .\/ Y ) )`
50 31 14 17 49 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) ) = ( X .\/ Y ) )`
51 1 2 3 5 oldmm4
` |-  ( ( K e. OL /\ X e. B /\ Z e. B ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) = ( X .\/ Z ) )`
52 31 14 20 51 syl3anc
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) = ( X .\/ Z ) )`
53 50 52 oveq12d
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) ) ./\ ( ( oc ` K ) ` ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) = ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) )`
54 48 53 eqtr2d
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Y /\ X C Z ) ) -> ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) = ( ( oc ` K ) ` ( ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Y ) ) .\/ ( ( ( oc ` K ) ` X ) ./\ ( ( oc ` K ) ` Z ) ) ) ) )`
` |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) /\ ( X C Y /\ X C Z ) ) -> ( X .\/ ( Y ./\ Z ) ) = ( ( X .\/ Y ) ./\ ( X .\/ Z ) ) )`