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Metamath Proof Explorer

Theorem lplni2

Description: The join of 3 different atoms is a lattice plane. (Contributed by NM, 4-Jul-2012)

Ref Expression
Hypotheses lplni2.l 
|- .<_ = ( le ` K )
lplni2.j 
|- .\/ = ( join ` K )
lplni2.a 
|- A = ( Atoms ` K )
lplni2.p 
|- P = ( LPlanes ` K )
Assertion lplni2 
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. P )

Proof

Step Hyp Ref Expression
1 lplni2.l 
 |-  .<_ = ( le ` K )
2 lplni2.j 
 |-  .\/ = ( join ` K )
3 lplni2.a 
 |-  A = ( Atoms ` K )
4 lplni2.p 
 |-  P = ( LPlanes ` K )
5 simp2 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( Q e. A /\ R e. A /\ S e. A ) )
6 simp3l 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> Q =/= R )
7 simp3r 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> -. S .<_ ( Q .\/ R ) )
8 eqidd 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ S ) )
9 neeq1 
 |-  ( q = Q -> ( q =/= r <-> Q =/= r ) )
10 oveq1 
 |-  ( q = Q -> ( q .\/ r ) = ( Q .\/ r ) )
11 10 breq2d 
 |-  ( q = Q -> ( s .<_ ( q .\/ r ) <-> s .<_ ( Q .\/ r ) ) )
12 11 notbid 
 |-  ( q = Q -> ( -. s .<_ ( q .\/ r ) <-> -. s .<_ ( Q .\/ r ) ) )
13 10 oveq1d 
 |-  ( q = Q -> ( ( q .\/ r ) .\/ s ) = ( ( Q .\/ r ) .\/ s ) )
14 13 eqeq2d 
 |-  ( q = Q -> ( ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) <-> ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ r ) .\/ s ) ) )
15 9 12 14 3anbi123d 
 |-  ( q = Q -> ( ( q =/= r /\ -. s .<_ ( q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) ) <-> ( Q =/= r /\ -. s .<_ ( Q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ r ) .\/ s ) ) ) )
16 neeq2 
 |-  ( r = R -> ( Q =/= r <-> Q =/= R ) )
17 oveq2 
 |-  ( r = R -> ( Q .\/ r ) = ( Q .\/ R ) )
18 17 breq2d 
 |-  ( r = R -> ( s .<_ ( Q .\/ r ) <-> s .<_ ( Q .\/ R ) ) )
19 18 notbid 
 |-  ( r = R -> ( -. s .<_ ( Q .\/ r ) <-> -. s .<_ ( Q .\/ R ) ) )
20 17 oveq1d 
 |-  ( r = R -> ( ( Q .\/ r ) .\/ s ) = ( ( Q .\/ R ) .\/ s ) )
21 20 eqeq2d 
 |-  ( r = R -> ( ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ r ) .\/ s ) <-> ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ s ) ) )
22 16 19 21 3anbi123d 
 |-  ( r = R -> ( ( Q =/= r /\ -. s .<_ ( Q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ r ) .\/ s ) ) <-> ( Q =/= R /\ -. s .<_ ( Q .\/ R ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ s ) ) ) )
23 breq1 
 |-  ( s = S -> ( s .<_ ( Q .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
24 23 notbid 
 |-  ( s = S -> ( -. s .<_ ( Q .\/ R ) <-> -. S .<_ ( Q .\/ R ) ) )
25 oveq2 
 |-  ( s = S -> ( ( Q .\/ R ) .\/ s ) = ( ( Q .\/ R ) .\/ S ) )
26 25 eqeq2d 
 |-  ( s = S -> ( ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ s ) <-> ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ S ) ) )
27 24 26 3anbi23d 
 |-  ( s = S -> ( ( Q =/= R /\ -. s .<_ ( Q .\/ R ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ s ) ) <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ S ) ) ) )
28 15 22 27 rspc3ev 
 |-  ( ( ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( Q .\/ R ) .\/ S ) ) ) -> E. q e. A E. r e. A E. s e. A ( q =/= r /\ -. s .<_ ( q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) ) )
29 5 6 7 8 28 syl13anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> E. q e. A E. r e. A E. s e. A ( q =/= r /\ -. s .<_ ( q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) ) )
30 simp1 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> K e. HL )
31 hllat 
 |-  ( K e. HL -> K e. Lat )
32 31 3ad2ant1 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> K e. Lat )
33 simp21 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> Q e. A )
34 simp22 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> R e. A )
35 eqid 
 |-  ( Base ` K ) = ( Base ` K )
36 35 2 3 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
37 30 33 34 36 syl3anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
38 simp23 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> S e. A )
39 35 3 atbase 
 |-  ( S e. A -> S e. ( Base ` K ) )
40 38 39 syl 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> S e. ( Base ` K ) )
41 35 2 latjcl 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) /\ S e. ( Base ` K ) ) -> ( ( Q .\/ R ) .\/ S ) e. ( Base ` K ) )
42 32 37 40 41 syl3anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. ( Base ` K ) )
43 35 1 2 3 4 islpln5 
 |-  ( ( K e. HL /\ ( ( Q .\/ R ) .\/ S ) e. ( Base ` K ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> E. q e. A E. r e. A E. s e. A ( q =/= r /\ -. s .<_ ( q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) ) ) )
44 30 42 43 syl2anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( ( Q .\/ R ) .\/ S ) e. P <-> E. q e. A E. r e. A E. s e. A ( q =/= r /\ -. s .<_ ( q .\/ r ) /\ ( ( Q .\/ R ) .\/ S ) = ( ( q .\/ r ) .\/ s ) ) ) )
45 29 44 mpbird 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( ( Q .\/ R ) .\/ S ) e. P )