Theorem cvrat 34236

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 27009 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 34235	. . 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 34178	. . . . . . . . 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 1003	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
10	1, 5	atbase 34104	. . . . . . . . 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 1004	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
13	1, 5	atbase 34104	. . . . . . . . 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 15546	. . . . . . . 8 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
16	8, 11, 14, 15	syl3anc 1228	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
17	16	breq2d 4459	. . . . . 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 1002	. . . . . 6 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
21	1, 2, 3, 4, 5	cvratlem 34235	. . . . . . 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 1230	. . . . 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 34180	. . . . . . . . 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 15538	. . . . . . . . 9 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
29	8, 11, 14, 28	syl3anc 1228	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
30		eqid 2467	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
31	1, 30, 2	pltnle 15453	. . . . . . . . 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 1228	. . . . . . 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 15549	. . . . . . . . 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 1230	. . . . . . . 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 973   = wceq 1379   e. wcel 1767   =/= wne 2662   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   lecple 14562   Posetcpo 15427   ltcplt 15428   joincjn 15431   0.cp0 15524   Latclat 15532   Atomscatm 34078   HLchlt 34165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-poset 15433  df-plt 15445  df-lub 15461  df-glb 15462  df-join 15463  df-meet 15464  df-p0 15526  df-lat 15533  df-clat 15595  df-oposet 33991  df-ol 33993  df-oml 33994  df-covers 34081  df-ats 34082  df-atl 34113  df-cvlat 34137  df-hlat 34166

This theorem is referenced by:  cvrat2  34243