Theorem cvrat 32707

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 27882 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 32706	. . 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 32649	. . . . . . . . 9 |- ( K e. HL -> K e. Lat )
8	7	adantr 466	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Lat )
9		simpr2 1012	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
10	1, 5	atbase 32575	. . . . . . . . 9 |- ( P e. A -> P e. B )
11	9, 10	syl 17	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. B )
12		simpr3 1013	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
13	1, 5	atbase 32575	. . . . . . . . 9 |- ( Q e. A -> Q e. B )
14	12, 13	syl 17	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. B )
15	1, 3	latjcom 16260	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
16	8, 11, 14, 15	syl3anc 1264	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17	16	breq2d 4438	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) <-> X .< ( Q .\/ P ) ) )
18	17	anbi2d 708	. . . . 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 458	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. HL )
20		simpr1 1011	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
21	1, 2, 3, 4, 5	cvratlem 32706	. . . . . . 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 435	. . . . . 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 1266	. . . . 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 218	. . . 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 430	. . 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 32651	. . . . . . . . 9 |- ( K e. HL -> K e. Poset )
27	26	adantr 466	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Poset )
28	1, 3	latjcl 16252	. . . . . . . . 9 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
29	8, 11, 14, 28	syl3anc 1264	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
30		eqid 2429	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
31	1, 30, 2	pltnle 16167	. . . . . . . . 9 |- ( ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) /\ X .< ( P .\/ Q ) ) -> -. ( P .\/ Q ) ( le ` K ) X )
32	31	ex 435	. . . . . . . 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 1264	. . . . . . 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 16263	. . . . . . . . 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 1266	. . . . . . . 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 210	. . . . . . 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 145	. . . . . 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 490	. . . . . 6 |- ( -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
39	37, 38	syl6ib 229	. . . . 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 430	. . . 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 720	. . 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 382	. 2 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> X e. A )
43	42	ex 435	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 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   Basecbs 15084   lecple 15160   Posetcpo 16140   ltcplt 16141   joincjn 16144   0.cp0 16238   Latclat 16246   Atomscatm 32549   HLchlt 32636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-preset 16128  df-poset 16146  df-plt 16159  df-lub 16175  df-glb 16176  df-join 16177  df-meet 16178  df-p0 16240  df-lat 16247  df-clat 16309  df-oposet 32462  df-ol 32464  df-oml 32465  df-covers 32552  df-ats 32553  df-atl 32584  df-cvlat 32608  df-hlat 32637

This theorem is referenced by:  cvrat2  32714