Theorem cvrat2 34715

Description: A Hilbert lattice element covered by the join of two distinct atoms is an atom. (atcvat2i 29246 analog.) (Contributed by NM, 30-Nov-2011.)

Hypotheses

Ref	Expression
cvrat2.b	|- B = ( Base ` K )
cvrat2.j	|- .\/ = ( join ` K )
cvrat2.c	|- C = ( <o ` K )
cvrat2.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cvrat2	|- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ ( P =/= Q /\ X C ( P .\/ Q ) ) ) -> X e. A )

Proof of Theorem cvrat2

Step	Hyp	Ref	Expression
1		cvrat2.b	. . . . . . . . 9 |- B = ( Base ` K )
2		cvrat2.j	. . . . . . . . 9 |- .\/ = ( join ` K )
3		eqid 2622	. . . . . . . . 9 |- ( 0. ` K ) = ( 0. ` K )
4		cvrat2.c	. . . . . . . . 9 |- C = ( <o ` K )
5		cvrat2.a	. . . . . . . . 9 |- A = ( Atoms ` K )
6	1, 2, 3, 4, 5	atcvrj0 34714	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ X C ( P .\/ Q ) ) -> ( X = ( 0. ` K ) <-> P = Q ) )
7	6	3expa 1265	. . . . . . 7 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X C ( P .\/ Q ) ) -> ( X = ( 0. ` K ) <-> P = Q ) )
8	7	necon3bid 2838	. . . . . 6 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X C ( P .\/ Q ) ) -> ( X =/= ( 0. ` K ) <-> P =/= Q ) )
9		simpl 473	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. HL )
10		simpr1 1067	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> X e. B )
11		hllat 34650	. . . . . . . . . . 11 |- ( K e. HL -> K e. Lat )
12	11	adantr 481	. . . . . . . . . 10 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> K e. Lat )
13		simpr2 1068	. . . . . . . . . . 11 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. A )
14	1, 5	atbase 34576	. . . . . . . . . . 11 |- ( P e. A -> P e. B )
15	13, 14	syl 17	. . . . . . . . . 10 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> P e. B )
16		simpr3 1069	. . . . . . . . . . 11 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. A )
17	1, 5	atbase 34576	. . . . . . . . . . 11 |- ( Q e. A -> Q e. B )
18	16, 17	syl 17	. . . . . . . . . 10 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> Q e. B )
19	1, 2	latjcl 17051	. . . . . . . . . 10 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) e. B )
20	12, 15, 18, 19	syl3anc 1326	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. B )
21		eqid 2622	. . . . . . . . . . 11 |- ( lt ` K ) = ( lt ` K )
22	1, 21, 4	cvrlt 34557	. . . . . . . . . 10 |- ( ( ( K e. HL /\ X e. B /\ ( P .\/ Q ) e. B ) /\ X C ( P .\/ Q ) ) -> X ( lt ` K ) ( P .\/ Q ) )
23	22	ex 450	. . . . . . . . 9 |- ( ( K e. HL /\ X e. B /\ ( P .\/ Q ) e. B ) -> ( X C ( P .\/ Q ) -> X ( lt ` K ) ( P .\/ Q ) ) )
24	9, 10, 20, 23	syl3anc 1326	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X C ( P .\/ Q ) -> X ( lt ` K ) ( P .\/ Q ) ) )
25	1, 21, 2, 3, 5	cvrat 34708	. . . . . . . . 9 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( X =/= ( 0. ` K ) /\ X ( lt ` K ) ( P .\/ Q ) ) -> X e. A ) )
26	25	expcomd 454	. . . . . . . 8 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X ( lt ` K ) ( P .\/ Q ) -> ( X =/= ( 0. ` K ) -> X e. A ) ) )
27	24, 26	syld 47	. . . . . . 7 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X C ( P .\/ Q ) -> ( X =/= ( 0. ` K ) -> X e. A ) ) )
28	27	imp 445	. . . . . 6 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X C ( P .\/ Q ) ) -> ( X =/= ( 0. ` K ) -> X e. A ) )
29	8, 28	sylbird 250	. . . . 5 |- ( ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) /\ X C ( P .\/ Q ) ) -> ( P =/= Q -> X e. A ) )
30	29	ex 450	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( X C ( P .\/ Q ) -> ( P =/= Q -> X e. A ) ) )
31	30	com23 86	. . 3 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( P =/= Q -> ( X C ( P .\/ Q ) -> X e. A ) ) )
32	31	impd 447	. 2 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) ) -> ( ( P =/= Q /\ X C ( P .\/ Q ) ) -> X e. A ) )
33	32	3impia 1261	1 |- ( ( K e. HL /\ ( X e. B /\ P e. A /\ Q e. A ) /\ ( P =/= Q /\ X C ( P .\/ Q ) ) ) -> X e. A )

Syntax hints:   -> 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   ltcplt 16941   joincjn 16944   0.cp0 17037   Latclat 17045   <o ccvr 34549   Atomscatm 34550   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:  cvrat3  34728  atcvrlln  34806  lncvrelatN  35067