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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 ) ) )
7 6 3adant3r3 
 |-  ( ( 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 ) ) )
9 8 3adant3r2 
 |-  ( ( 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 )
13 12 adantr 
 |-  ( ( 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 )
31 30 adantr 
 |-  ( ( K e. OML /\ ( X e. B /\ Y e. B /\ Z e. B ) ) -> K e. OL )
32 omllat 
 |-  ( K e. OML -> K e. Lat )
33 32 adantr 
 |-  ( ( 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 ) ) ) ) )
42 41 3adant3 
 |-  ( ( 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 ) ) ) ) )
55 54 3adant3 
 |-  ( ( 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 ) ) ) ) )
56 29 42 55 3eqtr4d 
 |-  ( ( 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 ) ) )