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Metamath Proof Explorer

Theorem cdlemg18c

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013)

Ref Expression
Hypotheses cdlemg12.l 
|- .<_ = ( le ` K )
cdlemg12.j 
|- .\/ = ( join ` K )
cdlemg12.m 
|- ./\ = ( meet ` K )
cdlemg12.a 
|- A = ( Atoms ` K )
cdlemg12.h 
|- H = ( LHyp ` K )
cdlemg12.t 
|- T = ( ( LTrn ` K ) ` W )
cdlemg12b.r 
|- R = ( ( trL ` K ) ` W )
cdlemg18b.u 
|- U = ( ( P .\/ Q ) ./\ W )
Assertion cdlemg18c 
|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` Q ) ) ./\ ( Q .\/ ( F ` P ) ) ) e. A )

Proof

Step Hyp Ref Expression
1 cdlemg12.l 
 |-  .<_ = ( le ` K )
2 cdlemg12.j 
 |-  .\/ = ( join ` K )
3 cdlemg12.m 
 |-  ./\ = ( meet ` K )
4 cdlemg12.a 
 |-  A = ( Atoms ` K )
5 cdlemg12.h 
 |-  H = ( LHyp ` K )
6 cdlemg12.t 
 |-  T = ( ( LTrn ` K ) ` W )
7 cdlemg12b.r 
 |-  R = ( ( trL ` K ) ` W )
8 cdlemg18b.u 
 |-  U = ( ( P .\/ Q ) ./\ W )
9 simp1l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> K e. HL )
10 simp21l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> P e. A )
11 simp1r 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> W e. H )
12 simp21 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P e. A /\ -. P .<_ W ) )
13 simp22l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> Q e. A )
14 simp31 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> P =/= Q )
15 1 2 3 4 5 8 cdleme0a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> U e. A )
16 9 11 12 13 14 15 syl212anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> U e. A )
17 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( K e. HL /\ W e. H ) )
18 simp23 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> F e. T )
19 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ Q e. A ) -> ( F ` Q ) e. A )
20 17 18 13 19 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( F ` Q ) e. A )
21 1 4 5 6 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )
22 17 18 10 21 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( F ` P ) e. A )
23 1 2 3 4 5 6 7 8 cdlemg18b 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> -. P .<_ ( U .\/ ( F ` Q ) ) )
24 simp32 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( F ` P ) =/= Q )
25 24 necomd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> Q =/= ( F ` P ) )
26 23 25 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( -. P .<_ ( U .\/ ( F ` Q ) ) /\ Q =/= ( F ` P ) ) )
27 simp33 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) )
28 1 2 3 4 5 6 7 cdlemg18a 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ F e. T ) /\ ( P =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) )
29 17 10 13 18 14 27 28 syl132anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) )
30 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q .<_ ( P .\/ Q ) )
31 9 10 13 30 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> Q .<_ ( P .\/ Q ) )
32 1 2 3 4 5 8 cdleme0cp 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
33 9 11 12 13 32 syl22anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( P .\/ U ) = ( P .\/ Q ) )
34 31 33 breqtrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> Q .<_ ( P .\/ U ) )
35 1 2 4 hlatlej2 
 |-  ( ( K e. HL /\ ( F ` Q ) e. A /\ ( F ` P ) e. A ) -> ( F ` P ) .<_ ( ( F ` Q ) .\/ ( F ` P ) ) )
36 9 20 22 35 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( F ` P ) .<_ ( ( F ` Q ) .\/ ( F ` P ) ) )
37 simp22 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( Q e. A /\ -. Q .<_ W ) )
38 5 6 1 2 4 3 8 cdlemg2kq 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ F e. T ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) )
39 17 12 37 18 38 syl121anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ U ) )
40 2 4 hlatjcom 
 |-  ( ( K e. HL /\ ( F ` P ) e. A /\ ( F ` Q ) e. A ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
41 9 22 20 40 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` P ) .\/ ( F ` Q ) ) = ( ( F ` Q ) .\/ ( F ` P ) ) )
42 2 4 hlatjcom 
 |-  ( ( K e. HL /\ ( F ` Q ) e. A /\ U e. A ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) )
43 9 20 16 42 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ U ) = ( U .\/ ( F ` Q ) ) )
44 39 41 43 3eqtr3d 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( F ` Q ) .\/ ( F ` P ) ) = ( U .\/ ( F ` Q ) ) )
45 36 44 breqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( F ` P ) .<_ ( U .\/ ( F ` Q ) ) )
46 34 45 jca 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( Q .<_ ( P .\/ U ) /\ ( F ` P ) .<_ ( U .\/ ( F ` Q ) ) ) )
47 1 2 3 4 ps-2c 
 |-  ( ( ( K e. HL /\ P e. A /\ U e. A ) /\ ( ( F ` Q ) e. A /\ Q e. A /\ ( F ` P ) e. A ) /\ ( ( -. P .<_ ( U .\/ ( F ` Q ) ) /\ Q =/= ( F ` P ) ) /\ ( P .\/ ( F ` Q ) ) =/= ( Q .\/ ( F ` P ) ) /\ ( Q .<_ ( P .\/ U ) /\ ( F ` P ) .<_ ( U .\/ ( F ` Q ) ) ) ) ) -> ( ( P .\/ ( F ` Q ) ) ./\ ( Q .\/ ( F ` P ) ) ) e. A )
48 9 10 16 20 13 22 26 29 46 47 syl333anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) /\ F e. T ) /\ ( P =/= Q /\ ( F ` P ) =/= Q /\ ( ( F ` Q ) .\/ ( F ` P ) ) =/= ( P .\/ Q ) ) ) -> ( ( P .\/ ( F ` Q ) ) ./\ ( Q .\/ ( F ` P ) ) ) e. A )