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Metamath Proof Explorer

Theorem isline4N

Description: Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses isline4.b 
|- B = ( Base ` K )
isline4.c 
|- C = (
isline4.a 
|- A = ( Atoms ` K )
isline4.n 
|- N = ( Lines ` K )
isline4.m 
|- M = ( pmap ` K )
Assertion isline4N 
|- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. p e. A p C X ) )

Proof

Step Hyp Ref Expression
1 isline4.b 
 |-  B = ( Base ` K )
2 isline4.c 
 |-  C = (
3 isline4.a 
 |-  A = ( Atoms ` K )
4 isline4.n 
 |-  N = ( Lines ` K )
5 isline4.m 
 |-  M = ( pmap ` K )
6 eqid 
 |-  ( join ` K ) = ( join ` K )
7 1 6 3 4 5 isline3 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. p e. A E. q e. A ( p =/= q /\ X = ( p ( join ` K ) q ) ) ) )
8 simpll 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> K e. HL )
9 1 3 atbase 
 |-  ( p e. A -> p e. B )
10 9 adantl 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> p e. B )
11 simplr 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> X e. B )
12 eqid 
 |-  ( le ` K ) = ( le ` K )
13 1 12 6 2 3 cvrval3 
 |-  ( ( K e. HL /\ p e. B /\ X e. B ) -> ( p C X <-> E. q e. A ( -. q ( le ` K ) p /\ ( p ( join ` K ) q ) = X ) ) )
14 8 10 11 13 syl3anc 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> ( p C X <-> E. q e. A ( -. q ( le ` K ) p /\ ( p ( join ` K ) q ) = X ) ) )
15 hlatl 
 |-  ( K e. HL -> K e. AtLat )
16 15 ad3antrrr 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> K e. AtLat )
17 simpr 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> q e. A )
18 simplr 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> p e. A )
19 12 3 atncmp 
 |-  ( ( K e. AtLat /\ q e. A /\ p e. A ) -> ( -. q ( le ` K ) p <-> q =/= p ) )
20 16 17 18 19 syl3anc 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> ( -. q ( le ` K ) p <-> q =/= p ) )
21 necom 
 |-  ( q =/= p <-> p =/= q )
22 20 21 bitrdi 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> ( -. q ( le ` K ) p <-> p =/= q ) )
23 eqcom 
 |-  ( ( p ( join ` K ) q ) = X <-> X = ( p ( join ` K ) q ) )
24 23 a1i 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> ( ( p ( join ` K ) q ) = X <-> X = ( p ( join ` K ) q ) ) )
25 22 24 anbi12d 
 |-  ( ( ( ( K e. HL /\ X e. B ) /\ p e. A ) /\ q e. A ) -> ( ( -. q ( le ` K ) p /\ ( p ( join ` K ) q ) = X ) <-> ( p =/= q /\ X = ( p ( join ` K ) q ) ) ) )
26 25 rexbidva 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> ( E. q e. A ( -. q ( le ` K ) p /\ ( p ( join ` K ) q ) = X ) <-> E. q e. A ( p =/= q /\ X = ( p ( join ` K ) q ) ) ) )
27 14 26 bitrd 
 |-  ( ( ( K e. HL /\ X e. B ) /\ p e. A ) -> ( p C X <-> E. q e. A ( p =/= q /\ X = ( p ( join ` K ) q ) ) ) )
28 27 rexbidva 
 |-  ( ( K e. HL /\ X e. B ) -> ( E. p e. A p C X <-> E. p e. A E. q e. A ( p =/= q /\ X = ( p ( join ` K ) q ) ) ) )
29 7 28 bitr4d 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. p e. A p C X ) )