Theorem cvrat 17062

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (Th. atcvati 11958 analog.)

Hypotheses

Ref	Expression
cvrat.b	|- B = (base` K)
cvrat.s	|- S = (lt` K)
cvrat.j	|- J = (join` K)
cvrat.z	|- Z = (0.` K)
cvrat.a	|- A = (AtomsNEW` K)

Assertion

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

Proof of Theorem cvrat

Step	Hyp	Ref	Expression
1		hlpos 17027	. . . . . . . . 9 |- (K e. HL -> K e. PosetNEW)
2	1	adantr 425	. . . . . . . 8 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> K e. PosetNEW)
3		simpr1 882	. . . . . . . 8 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> X e. B)
4		hllat 17026	. . . . . . . . . 10 |- (K e. HL -> K e. LatNEW)
5	4	adantr 425	. . . . . . . . 9 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> K e. LatNEW)
6		simpr2 883	. . . . . . . . . 10 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> P e. A)
7		cvrat.b	. . . . . . . . . . 11 |- B = (base` K)
8		cvrat.a	. . . . . . . . . . 11 |- A = (AtomsNEW` K)
9	7, 8	atombase 17003	. . . . . . . . . 10 |- (P e. A -> P e. B)
10	6, 9	syl 12	. . . . . . . . 9 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> P e. B)
11		simpr3 884	. . . . . . . . . 10 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> Q e. A)
12	7, 8	atombase 17003	. . . . . . . . . 10 |- (Q e. A -> Q e. B)
13	11, 12	syl 12	. . . . . . . . 9 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> Q e. B)
14		cvrat.j	. . . . . . . . . 10 |- J = (join` K)
15	7, 14	latjcl 16852	. . . . . . . . 9 |- ((K e. LatNEW /\ P e. B /\ Q e. B) -> (PJQ) e. B)
16	5, 10, 13, 15	syl111anc 1100	. . . . . . . 8 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (PJQ) e. B)
17		eqid 1884	. . . . . . . . . 10 |- (le` K) = (le` K)
18		cvrat.s	. . . . . . . . . 10 |- S = (lt` K)
19	7, 17, 18	pltnle 16786	. . . . . . . . 9 |- (((K e. PosetNEW /\ X e. B /\ (PJQ) e. B) /\ XS(PJQ)) -> -. (PJQ)(le` K)X)
20	19	ex 402	. . . . . . . 8 |- ((K e. PosetNEW /\ X e. B /\ (PJQ) e. B) -> (XS(PJQ) -> -. (PJQ)(le` K)X))
21	2, 3, 16, 20	syl111anc 1100	. . . . . . 7 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (XS(PJQ) -> -. (PJQ)(le` K)X))
22	7, 17, 14	latjle12 16863	. . . . . . . . 9 |- ((K e. LatNEW /\ (P e. B /\ Q e. B /\ X e. B)) -> ((P(le` K)X /\ Q(le` K)X) <-> (PJQ)(le` K)X))
23	5, 10, 13, 3, 22	syl13anc 1102	. . . . . . . 8 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((P(le` K)X /\ Q(le` K)X) <-> (PJQ)(le` K)X))
24	23	biimpd 170	. . . . . . 7 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((P(le` K)X /\ Q(le` K)X) -> (PJQ)(le` K)X))
25	21, 24	nsyld 132	. . . . . 6 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (XS(PJQ) -> -. (P(le` K)X /\ Q(le` K)X)))
26		ianor 329	. . . . . 6 |- (-. (P(le` K)X /\ Q(le` K)X) <-> (-. P(le` K)X \/ -. Q(le` K)X))
27	25, 26	syl6ib 229	. . . . 5 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (XS(PJQ) -> (-. P(le` K)X \/ -. Q(le` K)X)))
28	27	imp 377	. . . 4 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ XS(PJQ)) -> (-. P(le` K)X \/ -. Q(le` K)X))
29	28	adantrl 430	. . 3 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ (X =/= Z /\ XS(PJQ))) -> (-. P(le` K)X \/ -. Q(le` K)X))
30		cvrat.z	. . . . 5 |- Z = (0.` K)
31	7, 18, 14, 30, 8	cvratlem 17061	. . . 4 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ (X =/= Z /\ XS(PJQ))) -> (-. P(le` K)X -> X e. A))
32	7, 14	latjcom 16860	. . . . . . . . 9 |- ((K e. LatNEW /\ P e. B /\ Q e. B) -> (PJQ) = (QJP))
33	5, 10, 13, 32	syl111anc 1100	. . . . . . . 8 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (PJQ) = (QJP))
34	33	breq2d 3350	. . . . . . 7 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> (XS(PJQ) <-> XS(QJP)))
35	34	anbi2d 678	. . . . . 6 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((X =/= Z /\ XS(PJQ)) <-> (X =/= Z /\ XS(QJP))))
36		simpl 346	. . . . . . 7 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> K e. HL)
37	7, 18, 14, 30, 8	cvratlem 17061	. . . . . . . 8 |- (((K e. HL /\ (X e. B /\ Q e. A /\ P e. A)) /\ (X =/= Z /\ XS(QJP))) -> (-. Q(le` K)X -> X e. A))
38	37	ex 402	. . . . . . 7 |- ((K e. HL /\ (X e. B /\ Q e. A /\ P e. A)) -> ((X =/= Z /\ XS(QJP)) -> (-. Q(le` K)X -> X e. A)))
39	36, 3, 11, 6, 38	syl13anc 1102	. . . . . 6 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((X =/= Z /\ XS(QJP)) -> (-. Q(le` K)X -> X e. A)))
40	35, 39	sylbid 220	. . . . 5 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((X =/= Z /\ XS(PJQ)) -> (-. Q(le` K)X -> X e. A)))
41	40	imp 377	. . . 4 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ (X =/= Z /\ XS(PJQ))) -> (-. Q(le` K)X -> X e. A))
42	31, 41	jaod 469	. . 3 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ (X =/= Z /\ XS(PJQ))) -> ((-. P(le` K)X \/ -. Q(le` K)X) -> X e. A))
43	29, 42	mpd 29	. 2 |- (((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) /\ (X =/= Z /\ XS(PJQ))) -> X e. A)
44	43	ex 402	1 |- ((K e. HL /\ (X e. B /\ P e. A /\ Q e. A)) -> ((X =/= Z /\ XS(PJQ)) -> X e. A))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  PosetNEWcpo 16760  ltcplt 16761  joincjn 16766  0.cp0 16832  LatNEWclat 16834  AtomsNEWcatm 16981  HLchlt 16983

This theorem is referenced by:  cvrat2 17066

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-tru 1262  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-iota 5089  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-struct 16708  df-poset 16772  df-plt 16780  df-pge 16792  df-lub 16799  df-glb 16800  df-join 16801  df-meet 16802  df-p0 16841  df-lat 16847  df-clat 16848  df-oposet 16905  df-ol 16907  df-oml 16908  df-covers 16984  df-atoms 16985  df-atlat 16986  df-hlat 17017

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