Theorem atcvrlln 34806

Description: An element covering an atom is a lattice line and vice-versa. (Contributed by NM, 18-Jun-2012.)

Hypotheses

Ref	Expression
atcvrlln.b	|- B = ( Base ` K )
atcvrlln.c	|- C = ( <o ` K )
atcvrlln.a	|- A = ( Atoms ` K )
atcvrlln.n	|- N = ( LLines ` K )

Assertion

Ref	Expression
atcvrlln	|- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) -> ( X e. A <-> Y e. N ) )

Proof of Theorem atcvrlln

Dummy variables q p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpll1 1100	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ X e. A ) -> K e. HL )
2		simpll3 1102	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ X e. A ) -> Y e. B )
3		simpr 477	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ X e. A ) -> X e. A )
4		simplr 792	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ X e. A ) -> X C Y )
5		atcvrlln.b	. . . 4 |- B = ( Base ` K )
6		atcvrlln.c	. . . 4 |- C = ( <o ` K )
7		atcvrlln.a	. . . 4 |- A = ( Atoms ` K )
8		atcvrlln.n	. . . 4 |- N = ( LLines ` K )
9	5, 6, 7, 8	llni 34794	. . 3 |- ( ( ( K e. HL /\ Y e. B /\ X e. A ) /\ X C Y ) -> Y e. N )
10	1, 2, 3, 4, 9	syl31anc 1329	. 2 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ X e. A ) -> Y e. N )
11		simpr 477	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> Y e. N )
12		simpll1 1100	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> K e. HL )
13		simpll3 1102	. . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> Y e. B )
14		eqid 2622	. . . . . 6 |- ( join ` K ) = ( join ` K )
15	5, 14, 7, 8	islln3 34796	. . . . 5 |- ( ( K e. HL /\ Y e. B ) -> ( Y e. N <-> E. p e. A E. q e. A ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) )
16	12, 13, 15	syl2anc 693	. . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> ( Y e. N <-> E. p e. A E. q e. A ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) )
17	11, 16	mpbid 222	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> E. p e. A E. q e. A ( p =/= q /\ Y = ( p ( join ` K ) q ) ) )
18		simp1l1 1154	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> K e. HL )
19		simp1l2 1155	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> X e. B )
20		simp2l 1087	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> p e. A )
21		simp2r 1088	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> q e. A )
22		simp3l 1089	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> p =/= q )
23		simp1r 1086	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> X C Y )
24		simp3r 1090	. . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> Y = ( p ( join ` K ) q ) )
25	23, 24	breqtrd 4679	. . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> X C ( p ( join ` K ) q ) )
26	5, 14, 6, 7	cvrat2 34715	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ p e. A /\ q e. A ) /\ ( p =/= q /\ X C ( p ( join ` K ) q ) ) ) -> X e. A )
27	18, 19, 20, 21, 22, 25, 26	syl132anc 1344	. . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ ( p e. A /\ q e. A ) /\ ( p =/= q /\ Y = ( p ( join ` K ) q ) ) ) -> X e. A )
28	27	3exp 1264	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) -> ( ( p e. A /\ q e. A ) -> ( ( p =/= q /\ Y = ( p ( join ` K ) q ) ) -> X e. A ) ) )
29	28	rexlimdvv 3037	. . . 4 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) -> ( E. p e. A E. q e. A ( p =/= q /\ Y = ( p ( join ` K ) q ) ) -> X e. A ) )
30	29	adantr 481	. . 3 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> ( E. p e. A E. q e. A ( p =/= q /\ Y = ( p ( join ` K ) q ) ) -> X e. A ) )
31	17, 30	mpd 15	. 2 |- ( ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) /\ Y e. N ) -> X e. A )
32	10, 31	impbida 877	1 |- ( ( ( K e. HL /\ X e. B /\ Y e. B ) /\ X C Y ) -> ( X e. A <-> Y e. N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   joincjn 16944   <o ccvr 34549   Atomscatm 34550   HLchlt 34637   LLinesclln 34777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-llines 34784

This theorem is referenced by:  llncvrlpln  34844  2llnmj  34846  2llnm2N  34854