Theorem atltcvr 34721

Description: An equivalence of less-than ordering and covers relation. (Contributed by NM, 7-Feb-2012.)

Hypotheses

Ref	Expression
atltcvr.s	|- .< = ( lt ` K )
atltcvr.j	|- .\/ = ( join ` K )
atltcvr.a	|- A = ( Atoms ` K )
atltcvr.c	|- C = ( <o ` K )

Assertion

Ref	Expression
atltcvr	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) <-> P C ( Q .\/ R ) ) )

Proof of Theorem atltcvr

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . . 6 |- ( Q = R -> ( Q .\/ R ) = ( R .\/ R ) )
2		simpr3 1069	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
3		atltcvr.j	. . . . . . . 8 |- .\/ = ( join ` K )
4		atltcvr.a	. . . . . . . 8 |- A = ( Atoms ` K )
5	3, 4	hlatjidm 34655	. . . . . . 7 |- ( ( K e. HL /\ R e. A ) -> ( R .\/ R ) = R )
6	2, 5	syldan 487	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( R .\/ R ) = R )
7	1, 6	sylan9eqr 2678	. . . . 5 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( Q .\/ R ) = R )
8	7	breq2d 4665	. . . 4 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< ( Q .\/ R ) <-> P .< R ) )
9		hlatl 34647	. . . . . . . 8 |- ( K e. HL -> K e. AtLat )
10	9	adantr 481	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. AtLat )
11		simpr1 1067	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
12		atltcvr.s	. . . . . . . 8 |- .< = ( lt ` K )
13	12, 4	atnlt 34600	. . . . . . 7 |- ( ( K e. AtLat /\ P e. A /\ R e. A ) -> -. P .< R )
14	10, 11, 2, 13	syl3anc 1326	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> -. P .< R )
15	14	pm2.21d 118	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< R -> P C ( Q .\/ R ) ) )
16	15	adantr 481	. . . 4 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< R -> P C ( Q .\/ R ) ) )
17	8, 16	sylbid 230	. . 3 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q = R ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) )
18		simpl 473	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. HL )
19		hllat 34650	. . . . . . . 8 |- ( K e. HL -> K e. Lat )
20	19	adantr 481	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat )
21		simpr2 1068	. . . . . . . 8 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
22		eqid 2622	. . . . . . . . 9 |- ( Base ` K ) = ( Base ` K )
23	22, 4	atbase 34576	. . . . . . . 8 |- ( Q e. A -> Q e. ( Base ` K ) )
24	21, 23	syl 17	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. ( Base ` K ) )
25	22, 4	atbase 34576	. . . . . . . 8 |- ( R e. A -> R e. ( Base ` K ) )
26	2, 25	syl 17	. . . . . . 7 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. ( Base ` K ) )
27	22, 3	latjcl 17051	. . . . . . 7 |- ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
28	20, 24, 26, 27	syl3anc 1326	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( Q .\/ R ) e. ( Base ` K ) )
29		eqid 2622	. . . . . . 7 |- ( le ` K ) = ( le ` K )
30	29, 12	pltle 16961	. . . . . 6 |- ( ( K e. HL /\ P e. A /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) )
31	18, 11, 28, 30	syl3anc 1326	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) )
32	31	adantr 481	. . . 4 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P .< ( Q .\/ R ) -> P ( le ` K ) ( Q .\/ R ) ) )
33		simpll 790	. . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> K e. HL )
34		simplr 792	. . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( P e. A /\ Q e. A /\ R e. A ) )
35		simpr 477	. . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) )
36	33, 34, 35	3jca 1242	. . . . . . 7 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) )
37	36	anassrs 680	. . . . . 6 |- ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) /\ P ( le ` K ) ( Q .\/ R ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) )
38		atltcvr.c	. . . . . . 7 |- C = ( <o ` K )
39	29, 3, 38, 4	atcvrj2 34719	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( Q =/= R /\ P ( le ` K ) ( Q .\/ R ) ) ) -> P C ( Q .\/ R ) )
40	37, 39	syl 17	. . . . 5 |- ( ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) /\ P ( le ` K ) ( Q .\/ R ) ) -> P C ( Q .\/ R ) )
41	40	ex 450	. . . 4 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P ( le ` K ) ( Q .\/ R ) -> P C ( Q .\/ R ) ) )
42	32, 41	syld 47	. . 3 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ Q =/= R ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) )
43	17, 42	pm2.61dane 2881	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) -> P C ( Q .\/ R ) ) )
44	22, 4	atbase 34576	. . . 4 |- ( P e. A -> P e. ( Base ` K ) )
45	11, 44	syl 17	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. ( Base ` K ) )
46	22, 12, 38	cvrlt 34557	. . . 4 |- ( ( ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) /\ P C ( Q .\/ R ) ) -> P .< ( Q .\/ R ) )
47	46	ex 450	. . 3 |- ( ( K e. HL /\ P e. ( Base ` K ) /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( P C ( Q .\/ R ) -> P .< ( Q .\/ R ) ) )
48	18, 45, 28, 47	syl3anc 1326	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P C ( Q .\/ R ) -> P .< ( Q .\/ R ) ) )
49	43, 48	impbid 202	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .< ( Q .\/ R ) <-> P C ( Q .\/ R ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   ltcplt 16941   joincjn 16944   Latclat 17045   <o ccvr 34549   Atomscatm 34550   AtLatcal 34551   HLchlt 34637

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

This theorem is referenced by:  atlt  34723  2atlt  34725  atexchltN  34727