Theorem cvrat 32987

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 28039 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 32986	. . 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 32929	. . . . . . . . 9 |- ( K e. HL -> K e. Lat )
8	7	adantr 467	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Lat )
9		simpr2 1015	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
10	1, 5	atbase 32855	. . . . . . . . 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 1016	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
13	1, 5	atbase 32855	. . . . . . . . 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 16305	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
16	8, 11, 14, 15	syl3anc 1268	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17	16	breq2d 4414	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X .< ( P .\/ Q ) <-> X .< ( Q .\/ P ) ) )
18	17	anbi2d 710	. . . . 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 459	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. HL )
20		simpr1 1014	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
21	1, 2, 3, 4, 5	cvratlem 32986	. . . . . . 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 436	. . . . . 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 1270	. . . . 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 219	. . . 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 431	. . 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 32931	. . . . . . . . 9 |- ( K e. HL -> K e. Poset )
27	26	adantr 467	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Poset )
28	1, 3	latjcl 16297	. . . . . . . . 9 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
29	8, 11, 14, 28	syl3anc 1268	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
30		eqid 2451	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
31	1, 30, 2	pltnle 16212	. . . . . . . . 9 |- ( ( ( K e. Poset /\ X e. B /\ ( P .\/ Q ) e. B ) /\ X .< ( P .\/ Q ) ) -> -. ( P .\/ Q ) ( le ` K ) X )
32	31	ex 436	. . . . . . . 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 1268	. . . . . . 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 16308	. . . . . . . . 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 1270	. . . . . . . 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 211	. . . . . . 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 146	. . . . . 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 491	. . . . . 6 |- ( -. ( P ( le ` K ) X /\ Q ( le ` K ) X ) <-> ( -. P ( le ` K ) X \/ -. Q ( le ` K ) X ) )
39	37, 38	syl6ib 230	. . . . 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 431	. . . 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 722	. . 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 383	. 2 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ ( X =/= .0. /\ X .< ( P .\/ Q ) ) ) -> X e. A )
43	42	ex 436	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 188   \/ wo 370   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   class class class wbr 4402   ` cfv 5582  (class class class)co 6290   Basecbs 15121   lecple 15197   Posetcpo 16185   ltcplt 16186   joincjn 16189   0.cp0 16283   Latclat 16291   Atomscatm 32829   HLchlt 32916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-lat 16292  df-clat 16354  df-oposet 32742  df-ol 32744  df-oml 32745  df-covers 32832  df-ats 32833  df-atl 32864  df-cvlat 32888  df-hlat 32917

This theorem is referenced by:  cvrat2  32994