Metamath Proof Explorer

Theorem lncvrelatN

Description: A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lncvrelat.b 
|- B = ( Base ` K )
lncvrelat.c 
|- C = (
lncvrelat.a 
|- A = ( Atoms ` K )
lncvrelat.n 
|- N = ( Lines ` K )
lncvrelat.m 
|- M = ( pmap ` K )
Assertion lncvrelatN 
|- ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( ( M ` X ) e. N /\ P C X ) ) -> P e. A )

Proof

Step Hyp Ref Expression
1 lncvrelat.b 
 |-  B = ( Base ` K )
2 lncvrelat.c 
 |-  C = (
3 lncvrelat.a 
 |-  A = ( Atoms ` K )
4 lncvrelat.n 
 |-  N = ( Lines ` K )
5 lncvrelat.m 
 |-  M = ( pmap ` K )
6 hllat 
 |-  ( K e. HL -> K e. Lat )
7 6 3ad2ant1 
 |-  ( ( K e. HL /\ X e. B /\ P e. B ) -> K e. Lat )
8 eqid 
 |-  ( join ` K ) = ( join ` K )
9 8 3 4 5 isline2 
 |-  ( K e. Lat -> ( ( M ` X ) e. N <-> E. q e. A E. r e. A ( q =/= r /\ ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) ) ) )
10 7 9 syl 
 |-  ( ( K e. HL /\ X e. B /\ P e. B ) -> ( ( M ` X ) e. N <-> E. q e. A E. r e. A ( q =/= r /\ ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) ) ) )
11 simpll1 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> K e. HL )
12 simpll2 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> X e. B )
13 11 6 syl 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> K e. Lat )
14 simplrl 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> q e. A )
15 1 3 atbase 
 |-  ( q e. A -> q e. B )
16 14 15 syl 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> q e. B )
17 simplrr 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> r e. A )
18 1 3 atbase 
 |-  ( r e. A -> r e. B )
19 17 18 syl 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> r e. B )
20 1 8 latjcl 
 |-  ( ( K e. Lat /\ q e. B /\ r e. B ) -> ( q ( join ` K ) r ) e. B )
21 13 16 19 20 syl3anc 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( q ( join ` K ) r ) e. B )
22 1 5 pmap11 
 |-  ( ( K e. HL /\ X e. B /\ ( q ( join ` K ) r ) e. B ) -> ( ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) <-> X = ( q ( join ` K ) r ) ) )
23 11 12 21 22 syl3anc 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) <-> X = ( q ( join ` K ) r ) ) )
24 breq2 
 |-  ( X = ( q ( join ` K ) r ) -> ( P C X <-> P C ( q ( join ` K ) r ) ) )
25 24 biimpd 
 |-  ( X = ( q ( join ` K ) r ) -> ( P C X -> P C ( q ( join ` K ) r ) ) )
26 11 adantr 
 |-  ( ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) /\ P C ( q ( join ` K ) r ) ) -> K e. HL )
27 simpll3 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> P e. B )
28 27 14 17 3jca 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( P e. B /\ q e. A /\ r e. A ) )
29 28 adantr 
 |-  ( ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) /\ P C ( q ( join ` K ) r ) ) -> ( P e. B /\ q e. A /\ r e. A ) )
30 simplr 
 |-  ( ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) /\ P C ( q ( join ` K ) r ) ) -> q =/= r )
31 simpr 
 |-  ( ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) /\ P C ( q ( join ` K ) r ) ) -> P C ( q ( join ` K ) r ) )
32 1 8 2 3 cvrat2 
 |-  ( ( K e. HL /\ ( P e. B /\ q e. A /\ r e. A ) /\ ( q =/= r /\ P C ( q ( join ` K ) r ) ) ) -> P e. A )
33 26 29 30 31 32 syl112anc 
 |-  ( ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) /\ P C ( q ( join ` K ) r ) ) -> P e. A )
34 33 ex 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( P C ( q ( join ` K ) r ) -> P e. A ) )
35 25 34 syl9r 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( X = ( q ( join ` K ) r ) -> ( P C X -> P e. A ) ) )
36 23 35 sylbid 
 |-  ( ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) /\ q =/= r ) -> ( ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) -> ( P C X -> P e. A ) ) )
37 36 expimpd 
 |-  ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( q e. A /\ r e. A ) ) -> ( ( q =/= r /\ ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) ) -> ( P C X -> P e. A ) ) )
38 37 rexlimdvva 
 |-  ( ( K e. HL /\ X e. B /\ P e. B ) -> ( E. q e. A E. r e. A ( q =/= r /\ ( M ` X ) = ( M ` ( q ( join ` K ) r ) ) ) -> ( P C X -> P e. A ) ) )
39 10 38 sylbid 
 |-  ( ( K e. HL /\ X e. B /\ P e. B ) -> ( ( M ` X ) e. N -> ( P C X -> P e. A ) ) )
40 39 imp32 
 |-  ( ( ( K e. HL /\ X e. B /\ P e. B ) /\ ( ( M ` X ) e. N /\ P C X ) ) -> P e. A )