Theorem cvrat 33063

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 25788 analog.) (Contributed by NM, 22-Nov-2011.)

Hypotheses

Ref	Expression
cvrat.b	|- B = ( Base ` K )
cvrat.s	|- .< = ( lt ` K )
cvrat.j	|- .\/ = ( join ` K )
cvrat.z	|- .0. = ( 0. ` K )
cvrat.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cvrat	|- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> X e. A ) )

Proof of Theorem cvrat

Step	Hyp	Ref	Expression
1		cvrat.b	. . . 4 |- B = ( Base ` K )
2		cvrat.s	. . . 4 |- .< = ( lt ` K )
3		cvrat.j	. . . 4 |- .\/ = ( join ` K )
4		cvrat.z	. . . 4 |- .0. = ( 0. ` K )
5		cvrat.a	. . . 4 |- A = ( Atoms ` K )
6	1, 2, 3, 4, 5	cvratlem 33062	. . 3 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. P ( le ` K ) X -> X e. A ) )
7		hllat 33005	. . . . . . . . 9 |- ( K e. HL -> K e. Lat )
8	7	adantr 465	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Lat )
9		simpr2 995	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
10	1, 5	atbase 32931	. . . . . . . . 9 |- ( P e. A -> P e. B )
11	9, 10	syl 16	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. B )
12		simpr3 996	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
13	1, 5	atbase 32931	. . . . . . . . 9 |- ( Q e. A -> Q e. B )
14	12, 13	syl 16	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. B )
15	1, 3	latjcom 15227	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
16	8, 11, 14, 15	syl3anc 1218	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17	16	breq2d 4302	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) <-> X .< ( Q .\/ P ) ) )
18	17	anbi2d 703	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) <-> ( X =/= .0. /\ X .< ( Q .\/ P ) ) ) )
19		simpl 457	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. HL )
20		simpr1 994	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
21	1, 2, 3, 4, 5	cvratlem 33062	. . . . . . 7 |- ( ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) /\ ( X =/= .0. /\ X .< ( Q .\/ P ) ) ) -> ( -. Q ( le ` K ) X -> X e. A ) )
22	21	ex 434	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) -> ( ( X =/= .0. /\ X .< ( Q .\/ P ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
23	19, 20, 12, 9, 22	syl13anc 1220	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( Q .\/ P ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
24	18, 23	sylbid 215	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> ( -. Q ( le ` K ) X -> X e. A ) ) )
25	24	imp 429	. . 3 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. Q ( le ` K ) X -> X e. A ) )
26		hlpos 33007	. . . . . . . . 9 |- ( K e. HL -> K e. Poset )
27	26	adantr 465	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Poset )
28	1, 3	latjcl 15219	. . . . . . . . 9 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
29	8, 11, 14, 28	syl3anc 1218	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
30		eqid 2441	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
31	1, 30, 2	pltnle 15134	. . . . . . . . 9 |- ( ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) /\ X .< ( P .\/ Q ) ) -> -. ( P .\/ Q ) ( le ` K ) X )
32	31	ex 434	. . . . . . . 8 |- ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) -> ( X .< ( P .\/ Q ) -> -. ( P .\/ Q ) ( le ` K ) X ) )
33	27, 20, 29, 32	syl3anc 1218	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> -. ( P .\/ Q ) ( le ` K ) X ) )
34	1, 30, 3	latjle12 15230	. . . . . . . . 9 |- ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( P .\/ Q ) ( le ` K ) X ) )
35	8, 11, 14, 20, 34	syl13anc 1220	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( P .\/ Q ) ( le ` K ) X ) )
36	35	biimpd 207	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P ( le ` K ) X /\ Q ( le ` K ) X ) -> ( P .\/ Q ) ( le ` K ) X ) )
37	33, 36	nsyld 140	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) ) )
38		ianor 488	. . . . . 6 |- ( -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
39	37, 38	syl6ib 226	. . . . 5 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) ) )
40	39	imp 429	. . . 4 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X .< ( P .\/ Q ) ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
41	40	adantrl 715	. . 3 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
42	6, 25, 41	mpjaod 381	. 2 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> X e. A )
43	42	ex 434	1 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= .0. /\ X .< ( P .\/ Q ) ) -> X e. A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2604   class class class wbr 4290   ` cfv 5416  (class class class)co 6089   Basecbs 14172   lecple 14243   Posetcpo 15108   ltcplt 15109   joincjn 15112   0.cp0 15205   Latclat 15213   Atomscatm 32905   HLchlt 32992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-poset 15114  df-plt 15126  df-lub 15142  df-glb 15143  df-join 15144  df-meet 15145  df-p0 15207  df-lat 15214  df-clat 15276  df-oposet 32818  df-ol 32820  df-oml 32821  df-covers 32908  df-ats 32909  df-atl 32940  df-cvlat 32964  df-hlat 32993

This theorem is referenced by:  cvrat2  33070