Theorem hlrelat 17051

Description: A Hilbert lattice is relatively atomic. Remark 2 of [Kalmbach] p. 149. (Th. chrelati 11936 analog.)

Hypotheses

Ref	Expression
hlrelat5.b	|- B = (base` K)
hlrelat5.l	|- L = (le` K)
hlrelat5.s	|- S = (lt` K)
hlrelat5.j	|- J = (join` K)
hlrelat5.a	|- A = (AtomsNEW` K)

Assertion

Ref	Expression
hlrelat	|- (((K e. HL /\ X e. B /\ Y e. B) /\ XSY) -> E.p e. A (XS(XJp) /\ (XJp)LY))

Distinct variable groups:   A,p   B,p   K,p   L,p   X,p   Y,p   S,p

Proof of Theorem hlrelat

Step	Hyp	Ref	Expression
1		hlrelat5.b	. . . 4 |- B = (base` K)
2		hlrelat5.l	. . . 4 |- L = (le` K)
3		hlrelat5.s	. . . 4 |- S = (lt` K)
4		hlrelat5.a	. . . 4 |- A = (AtomsNEW` K)
5	1, 2, 3, 4	hlrelat1 17049	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XSY -> E.p e. A (-. pLX /\ pLY)))
6	5	imp 377	. 2 |- (((K e. HL /\ X e. B /\ Y e. B) /\ XSY) -> E.p e. A (-. pLX /\ pLY))
7		simpll1 915	. . . . . 6 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> K e. HL)
8		hllat 17026	. . . . . 6 |- (K e. HL -> K e. LatNEW)
9	7, 8	syl 12	. . . . 5 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> K e. LatNEW)
10		simpll2 916	. . . . 5 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> X e. B)
11	1, 4	atombase 17003	. . . . . 6 |- (p e. A -> p e. B)
12	11	adantl 424	. . . . 5 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> p e. B)
13		hlrelat5.j	. . . . . 6 |- J = (join` K)
14	1, 2, 3, 13	latnle 16880	. . . . 5 |- ((K e. LatNEW /\ X e. B /\ p e. B) -> (-. pLX <-> XS(XJp)))
15	9, 10, 12, 14	syl111anc 1100	. . . 4 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> (-. pLX <-> XS(XJp)))
16	2, 3	pltle 16782	. . . . . . . 8 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XSY -> XLY))
17	16	imp 377	. . . . . . 7 |- (((K e. HL /\ X e. B /\ Y e. B) /\ XSY) -> XLY)
18	17	adantr 425	. . . . . 6 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> XLY)
19	18	biantrurd 796	. . . . 5 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> (pLY <-> (XLY /\ pLY)))
20		simpll3 917	. . . . . 6 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> Y e. B)
21	1, 2, 13	latjle12 16863	. . . . . 6 |- ((K e. LatNEW /\ (X e. B /\ p e. B /\ Y e. B)) -> ((XLY /\ pLY) <-> (XJp)LY))
22	9, 10, 12, 20, 21	syl13anc 1102	. . . . 5 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> ((XLY /\ pLY) <-> (XJp)LY))
23	19, 22	bitrd 587	. . . 4 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> (pLY <-> (XJp)LY))
24	15, 23	anbi12d 690	. . 3 |- ((((K e. HL /\ X e. B /\ Y e. B) /\ XSY) /\ p e. A) -> ((-. pLX /\ pLY) <-> (XS(XJp) /\ (XJp)LY)))
25	24	rexbidva 2120	. 2 |- (((K e. HL /\ X e. B /\ Y e. B) /\ XSY) -> (E.p e. A (-. pLX /\ pLY) <-> E.p e. A (XS(XJp) /\ (XJp)LY)))
26	6, 25	mpbid 212	1 |- (((K e. HL /\ X e. B /\ Y e. B) /\ XSY) -> E.p e. A (XS(XJp) /\ (XJp)LY))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  ltcplt 16761  joincjn 16766  LatNEWclat 16834  AtomsNEWcatm 16981  HLchlt 16983

This theorem is referenced by:  hlrelat2 17052

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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