Theorem hlrelat2 17052

Description: A consequence of relative atomicity. (Th. chrelat2i 11937 analog.)

Hypotheses

Ref	Expression
hlrelat2.b	|- B = (base` K)
hlrelat2.l	|- L = (le` K)
hlrelat2.a	|- A = (AtomsNEW` K)

Assertion

Ref	Expression
hlrelat2	|- ((K e. HL /\ X e. B /\ Y e. B) -> (-. XLY <-> E.p e. A (pLX /\ -. pLY)))

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

Proof of Theorem hlrelat2

Step	Hyp	Ref	Expression
1		hlrelat2.b	. . . . 5 |- B = (base` K)
2		hlrelat2.l	. . . . 5 |- L = (le` K)
3		eqid 1884	. . . . 5 |- (lt` K) = (lt` K)
4		eqid 1884	. . . . 5 |- (meet` K) = (meet` K)
5	1, 2, 3, 4	latnlemlt 16879	. . . 4 |- ((K e. LatNEW /\ X e. B /\ Y e. B) -> (-. XLY <-> (X(meet` K)Y)(lt` K)X))
6		hllat 17026	. . . 4 |- (K e. HL -> K e. LatNEW)
7	5, 6	syl3an1 1130	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (-. XLY <-> (X(meet` K)Y)(lt` K)X))
8		simp1 876	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> K e. HL)
9	1, 4	latmcl 16853	. . . . . 6 |- ((K e. LatNEW /\ X e. B /\ Y e. B) -> (X(meet` K)Y) e. B)
10	9, 6	syl3an1 1130	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> (X(meet` K)Y) e. B)
11		simp2 877	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> X e. B)
12		eqid 1884	. . . . . . 7 |- (join` K) = (join` K)
13		hlrelat2.a	. . . . . . 7 |- A = (AtomsNEW` K)
14	1, 2, 3, 12, 13	hlrelat 17051	. . . . . 6 |- (((K e. HL /\ (X(meet` K)Y) e. B /\ X e. B) /\ (X(meet` K)Y)(lt` K)X) -> E.p e. A ((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX))
15	14	ex 402	. . . . 5 |- ((K e. HL /\ (X(meet` K)Y) e. B /\ X e. B) -> ((X(meet` K)Y)(lt` K)X -> E.p e. A ((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX)))
16	8, 10, 11, 15	syl111anc 1100	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> ((X(meet` K)Y)(lt` K)X -> E.p e. A ((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX)))
17		simpl1 879	. . . . . . . . . 10 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> K e. HL)
18	17, 6	syl 12	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> K e. LatNEW)
19	10	adantr 425	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (X(meet` K)Y) e. B)
20	1, 13	atombase 17003	. . . . . . . . . 10 |- (p e. A -> p e. B)
21	20	adantl 424	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> p e. B)
22		simpl2 880	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> X e. B)
23	1, 2, 12	latjle12 16863	. . . . . . . . 9 |- ((K e. LatNEW /\ ((X(meet` K)Y) e. B /\ p e. B /\ X e. B)) -> (((X(meet` K)Y)LX /\ pLX) <-> ((X(meet` K)Y)(join` K)p)LX))
24	18, 19, 21, 22, 23	syl13anc 1102	. . . . . . . 8 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (((X(meet` K)Y)LX /\ pLX) <-> ((X(meet` K)Y)(join` K)p)LX))
25		simpr 350	. . . . . . . 8 |- (((X(meet` K)Y)LX /\ pLX) -> pLX)
26	24, 25	syl6bir 232	. . . . . . 7 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (((X(meet` K)Y)(join` K)p)LX -> pLX))
27	26	adantld 426	. . . . . 6 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX) -> pLX))
28		simpl3 881	. . . . . . . . . . 11 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> Y e. B)
29	1, 2, 4	latlem12 16873	. . . . . . . . . . 11 |- ((K e. LatNEW /\ (p e. B /\ X e. B /\ Y e. B)) -> ((pLX /\ pLY) <-> pL(X(meet` K)Y)))
30	18, 21, 22, 28, 29	syl13anc 1102	. . . . . . . . . 10 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> ((pLX /\ pLY) <-> pL(X(meet` K)Y)))
31	30	notbid 673	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (-. (pLX /\ pLY) <-> -. pL(X(meet` K)Y)))
32	1, 2, 3, 12	latnle 16880	. . . . . . . . . 10 |- ((K e. LatNEW /\ (X(meet` K)Y) e. B /\ p e. B) -> (-. pL(X(meet` K)Y) <-> (X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p)))
33	18, 19, 21, 32	syl111anc 1100	. . . . . . . . 9 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (-. pL(X(meet` K)Y) <-> (X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p)))
34	31, 33	bitrd 587	. . . . . . . 8 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (-. (pLX /\ pLY) <-> (X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p)))
35	34, 24	anbi12d 690	. . . . . . 7 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> ((-. (pLX /\ pLY) /\ ((X(meet` K)Y)LX /\ pLX)) <-> ((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX)))
36		pm3.21 306	. . . . . . . . . 10 |- (pLY -> (pLX -> (pLX /\ pLY)))
37		ianor 329	. . . . . . . . . . 11 |- (-. (-. (pLX /\ pLY) /\ pLX) <-> (-. -. (pLX /\ pLY) \/ -. pLX))
38		notnot 178	. . . . . . . . . . . 12 |- ((pLX /\ pLY) <-> -. -. (pLX /\ pLY))
39	38	orbi1i 276	. . . . . . . . . . 11 |- (((pLX /\ pLY) \/ -. pLX) <-> (-. -. (pLX /\ pLY) \/ -. pLX))
40		orcom 266	. . . . . . . . . . . 12 |- (((pLX /\ pLY) \/ -. pLX) <-> (-. pLX \/ (pLX /\ pLY)))
41		imor 251	. . . . . . . . . . . 12 |- ((pLX -> (pLX /\ pLY)) <-> (-. pLX \/ (pLX /\ pLY)))
42	40, 41	bitr4i 193	. . . . . . . . . . 11 |- (((pLX /\ pLY) \/ -. pLX) <-> (pLX -> (pLX /\ pLY)))
43	37, 39, 42	3bitr2ri 197	. . . . . . . . . 10 |- ((pLX -> (pLX /\ pLY)) <-> -. (-. (pLX /\ pLY) /\ pLX))
44	36, 43	sylib 215	. . . . . . . . 9 |- (pLY -> -. (-. (pLX /\ pLY) /\ pLX))
45	44	con2i 113	. . . . . . . 8 |- ((-. (pLX /\ pLY) /\ pLX) -> -. pLY)
46	45	adantrl 430	. . . . . . 7 |- ((-. (pLX /\ pLY) /\ ((X(meet` K)Y)LX /\ pLX)) -> -. pLY)
47	35, 46	syl6bir 232	. . . . . 6 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX) -> -. pLY))
48	27, 47	jcad 661	. . . . 5 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> (((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX) -> (pLX /\ -. pLY)))
49	48	reximdva 2203	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> (E.p e. A ((X(meet` K)Y)(lt` K)((X(meet` K)Y)(join` K)p) /\ ((X(meet` K)Y)(join` K)p)LX) -> E.p e. A (pLX /\ -. pLY)))
50	16, 49	syld 30	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> ((X(meet` K)Y)(lt` K)X -> E.p e. A (pLX /\ -. pLY)))
51	7, 50	sylbid 220	. 2 |- ((K e. HL /\ X e. B /\ Y e. B) -> (-. XLY -> E.p e. A (pLX /\ -. pLY)))
52	1, 2	lattr 16858	. . . . . . . . 9 |- ((K e. LatNEW /\ (p e. B /\ X e. B /\ Y e. B)) -> ((pLX /\ XLY) -> pLY))
53	18, 21, 22, 28, 52	syl13anc 1102	. . . . . . . 8 |- (((K e. HL /\ X e. B /\ Y e. B) /\ p e. A) -> ((pLX /\ XLY) -> pLY))
54	53	exp4b 410	. . . . . . 7 |- ((K e. HL /\ X e. B /\ Y e. B) -> (p e. A -> (pLX -> (XLY -> pLY))))
55	54	com34 40	. . . . . 6 |- ((K e. HL /\ X e. B /\ Y e. B) -> (p e. A -> (XLY -> (pLX -> pLY))))
56	55	com23 36	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY -> (p e. A -> (pLX -> pLY))))
57	56	r19.21adv 2181	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY -> A.p e. A (pLX -> pLY)))
58		iman 256	. . . . . 6 |- ((pLX -> pLY) <-> -. (pLX /\ -. pLY))
59	58	ralbii 2127	. . . . 5 |- (A.p e. A (pLX -> pLY) <-> A.p e. A -. (pLX /\ -. pLY))
60		ralnex 2113	. . . . 5 |- (A.p e. A -. (pLX /\ -. pLY) <-> -. E.p e. A (pLX /\ -. pLY))
61	59, 60	bitri 190	. . . 4 |- (A.p e. A (pLX -> pLY) <-> -. E.p e. A (pLX /\ -. pLY))
62	57, 61	syl6ib 229	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY -> -. E.p e. A (pLX /\ -. pLY)))
63	62	con2d 107	. 2 |- ((K e. HL /\ X e. B /\ Y e. B) -> (E.p e. A (pLX /\ -. pLY) -> -. XLY))
64	51, 63	impbid 574	1 |- ((K e. HL /\ X e. B /\ Y e. B) -> (-. XLY <-> E.p e. A (pLX /\ -. pLY)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  ltcplt 16761  joincjn 16766  meetcmee 16767  LatNEWclat 16834  AtomsNEWcatm 16981  HLchlt 16983

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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