Theorem hlatmstc 17047

Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (Th. hatomistici 11934 analog.)

Hypotheses

Ref	Expression
hlatmstc.b	|- B = (base` K)
hlatmstc.l	|- L = (le` K)
hlatmstc.u	|- U = (lub` K)
hlatmstc.a	|- A = (AtomsNEW` K)

Assertion

Ref	Expression
hlatmstc	|- ((K e. HL /\ X e. B) -> (U` {y e. A | yLX}) = X)

Distinct variable groups:   y,A   y,B   y,L   y,X

Proof of Theorem hlatmstc

Step	Hyp	Ref	Expression
1		hlclat 17022	. . . . 5 |- (K e. HL -> K e. CLat)
2		ssrab2 2692	. . . . . 6 |- {y e. B | yLX} C_ B
3	2	a1i 8	. . . . 5 |- (K e. HL -> {y e. B | yLX} C_ B)
4		hlatmstc.b	. . . . . . . 8 |- B = (base` K)
5		hlatmstc.a	. . . . . . . 8 |- A = (AtomsNEW` K)
6	4, 5	atomssbase 17004	. . . . . . 7 |- A C_ B
7		rabss2 2689	. . . . . . 7 |- (A C_ B -> {y e. A | yLX} C_ {y e. B | yLX})
8	6, 7	ax-mp 7	. . . . . 6 |- {y e. A | yLX} C_ {y e. B | yLX}
9	8	a1i 8	. . . . 5 |- (K e. HL -> {y e. A | yLX} C_ {y e. B | yLX})
10		hlatmstc.l	. . . . . 6 |- L = (le` K)
11		hlatmstc.u	. . . . . 6 |- U = (lub` K)
12	4, 10, 11	lubss 16897	. . . . 5 |- ((K e. CLat /\ {y e. B | yLX} C_ B /\ {y e. A | yLX} C_ {y e. B | yLX}) -> (U` {y e. A | yLX})L(U` {y e. B | yLX}))
13	1, 3, 9, 12	syl111anc 1100	. . . 4 |- (K e. HL -> (U` {y e. A | yLX})L(U` {y e. B | yLX}))
14	13	adantr 425	. . 3 |- ((K e. HL /\ X e. B) -> (U` {y e. A | yLX})L(U` {y e. B | yLX}))
15	4, 10, 11	lubid 16807	. . . 4 |- ((K e. PosetNEW /\ X e. B) -> (U` {y e. B | yLX}) = X)
16		hloml 17021	. . . . 5 |- (K e. HL -> K e. OML)
17		omlol 16961	. . . . . 6 |- (K e. OML -> K e. OL)
18		olpos 16942	. . . . . 6 |- (K e. OL -> K e. PosetNEW)
19	17, 18	syl 12	. . . . 5 |- (K e. OML -> K e. PosetNEW)
20	16, 19	syl 12	. . . 4 |- (K e. HL -> K e. PosetNEW)
21	15, 20	sylan 497	. . 3 |- ((K e. HL /\ X e. B) -> (U` {y e. B | yLX}) = X)
22	14, 21	breqtrd 3361	. 2 |- ((K e. HL /\ X e. B) -> (U` {y e. A | yLX})LX)
23	1	ad2antrr 440	. . . . . . . . . . 11 |- (((K e. HL /\ X e. B) /\ x e. {y e. A | yLX}) -> K e. CLat)
24		simpr 350	. . . . . . . . . . 11 |- (((K e. HL /\ X e. B) /\ x e. {y e. A | yLX}) -> x e. {y e. A | yLX})
25		ssrab2 2692	. . . . . . . . . . . . 13 |- {y e. A | yLX} C_ A
26	25, 6	sstri 2626	. . . . . . . . . . . 12 |- {y e. A | yLX} C_ B
27	26	a1i 8	. . . . . . . . . . 11 |- (((K e. HL /\ X e. B) /\ x e. {y e. A | yLX}) -> {y e. A | yLX} C_ B)
28	4, 10, 11	lubel 16898	. . . . . . . . . . 11 |- ((K e. CLat /\ x e. {y e. A | yLX} /\ {y e. A | yLX} C_ B) -> xL(U` {y e. A | yLX}))
29	23, 24, 27, 28	syl111anc 1100	. . . . . . . . . 10 |- (((K e. HL /\ X e. B) /\ x e. {y e. A | yLX}) -> xL(U` {y e. A | yLX}))
30	29	ex 402	. . . . . . . . 9 |- ((K e. HL /\ X e. B) -> (x e. {y e. A | yLX} -> xL(U` {y e. A | yLX})))
31		breq1 3341	. . . . . . . . . 10 |- (y = x -> (yLX <-> xLX))
32	31	elrab 2414	. . . . . . . . 9 |- (x e. {y e. A | yLX} <-> (x e. A /\ xLX))
33	30, 32	syl5ibr 224	. . . . . . . 8 |- ((K e. HL /\ X e. B) -> ((x e. A /\ xLX) -> xL(U` {y e. A | yLX})))
34	33	expdimp 406	. . . . . . 7 |- (((K e. HL /\ X e. B) /\ x e. A) -> (xLX -> xL(U` {y e. A | yLX})))
35		eqid 1884	. . . . . . . . . . . . 13 |- (0.` K) = (0.` K)
36	35, 5	atomn0 17006	. . . . . . . . . . . 12 |- ((K e. OP /\ x e. A) -> x =/= (0.` K))
37		hlop 17025	. . . . . . . . . . . 12 |- (K e. HL -> K e. OP)
38	36, 37	sylan 497	. . . . . . . . . . 11 |- ((K e. HL /\ x e. A) -> x =/= (0.` K))
39	38	adantlr 429	. . . . . . . . . 10 |- (((K e. HL /\ X e. B) /\ x e. A) -> x =/= (0.` K))
40	39	adantr 425	. . . . . . . . 9 |- ((((K e. HL /\ X e. B) /\ x e. A) /\ xL(U` {y e. A | yLX})) -> x =/= (0.` K))
41		hllat 17026	. . . . . . . . . . . . . . 15 |- (K e. HL -> K e. LatNEW)
42	41	ad2antrr 440	. . . . . . . . . . . . . 14 |- (((K e. HL /\ X e. B) /\ x e. A) -> K e. LatNEW)
43	4, 5	atombase 17003	. . . . . . . . . . . . . . 15 |- (x e. A -> x e. B)
44	43	adantl 424	. . . . . . . . . . . . . 14 |- (((K e. HL /\ X e. B) /\ x e. A) -> x e. B)
45	4, 11	clatlubcl 16890	. . . . . . . . . . . . . . . 16 |- ((K e. CLat /\ {y e. A | yLX} C_ B) -> (U` {y e. A | yLX}) e. B)
46	45, 1, 26	sylancl 525	. . . . . . . . . . . . . . 15 |- (K e. HL -> (U` {y e. A | yLX}) e. B)
47	46	ad2antrr 440	. . . . . . . . . . . . . 14 |- (((K e. HL /\ X e. B) /\ x e. A) -> (U` {y e. A | yLX}) e. B)
48		eqid 1884	. . . . . . . . . . . . . . . . 17 |- (oc` K) = (oc` K)
49	4, 48	opoccl 16921	. . . . . . . . . . . . . . . 16 |- ((K e. OP /\ (U` {y e. A | yLX}) e. B) -> ((oc` K)` (U` {y e. A | yLX})) e. B)
50	37, 46, 49	syl11anc 524	. . . . . . . . . . . . . . 15 |- (K e. HL -> ((oc` K)` (U` {y e. A | yLX})) e. B)
51	50	ad2antrr 440	. . . . . . . . . . . . . 14 |- (((K e. HL /\ X e. B) /\ x e. A) -> ((oc` K)` (U` {y e. A | yLX})) e. B)
52		eqid 1884	. . . . . . . . . . . . . . 15 |- (meet` K) = (meet` K)
53	4, 10, 52	latlem12 16873	. . . . . . . . . . . . . 14 |- ((K e. LatNEW /\ (x e. B /\ (U` {y e. A | yLX}) e. B /\ ((oc` K)` (U` {y e. A | yLX})) e. B)) -> ((xL(U` {y e. A | yLX}) /\ xL((oc` K)` (U` {y e. A | yLX}))) <-> xL((U` {y e. A | yLX})(meet` K)((oc` K)` (U` {y e. A | yLX})))))
54	42, 44, 47, 51, 53	syl13anc 1102	. . . . . . . . . . . . 13 |- (((K e. HL /\ X e. B) /\ x e. A) -> ((xL(U` {y e. A | yLX}) /\ xL((oc` K)` (U` {y e. A | yLX}))) <-> xL((U` {y e. A | yLX})(meet` K)((oc` K)` (U` {y e. A | yLX})))))
55	37	ad2antrr 440	. . . . . . . . . . . . . . 15 |- (((K e. HL /\ X e. B) /\ x e. A) -> K e. OP)
56	4, 48, 52, 35	opnoncon 16935	. . . . . . . . . . . . . . 15 |- ((K e. OP /\ (U` {y e. A | yLX}) e. B) -> ((U` {y e. A | yLX})(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K))
57	55, 47, 56	syl11anc 524	. . . . . . . . . . . . . 14 |- (((K e. HL /\ X e. B) /\ x e. A) -> ((U` {y e. A | yLX})(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K))
58	57	breq2d 3350	. . . . . . . . . . . . 13 |- (((K e. HL /\ X e. B) /\ x e. A) -> (xL((U` {y e. A | yLX})(meet` K)((oc` K)` (U` {y e. A | yLX}))) <-> xL(0.` K)))
59	4, 10, 35	ople0 16917	. . . . . . . . . . . . . 14 |- ((K e. OP /\ x e. B) -> (xL(0.` K) <-> x = (0.` K)))
60	37	adantr 425	. . . . . . . . . . . . . 14 |- ((K e. HL /\ X e. B) -> K e. OP)
61	59, 60, 43	syl2an 503	. . . . . . . . . . . . 13 |- (((K e. HL /\ X e. B) /\ x e. A) -> (xL(0.` K) <-> x = (0.` K)))
62	54, 58, 61	3bitrd 603	. . . . . . . . . . . 12 |- (((K e. HL /\ X e. B) /\ x e. A) -> ((xL(U` {y e. A | yLX}) /\ xL((oc` K)` (U` {y e. A | yLX}))) <-> x = (0.` K)))
63	62	biimpa 460	. . . . . . . . . . 11 |- ((((K e. HL /\ X e. B) /\ x e. A) /\ (xL(U` {y e. A | yLX}) /\ xL((oc` K)` (U` {y e. A | yLX})))) -> x = (0.` K))
64	63	expr 418	. . . . . . . . . 10 |- ((((K e. HL /\ X e. B) /\ x e. A) /\ xL(U` {y e. A | yLX})) -> (xL((oc` K)` (U` {y e. A | yLX})) -> x = (0.` K)))
65	64	necon3ad 2037	. . . . . . . . 9 |- ((((K e. HL /\ X e. B) /\ x e. A) /\ xL(U` {y e. A | yLX})) -> (x =/= (0.` K) -> -. xL((oc` K)` (U` {y e. A | yLX}))))
66	40, 65	mpd 29	. . . . . . . 8 |- ((((K e. HL /\ X e. B) /\ x e. A) /\ xL(U` {y e. A | yLX})) -> -. xL((oc` K)` (U` {y e. A | yLX})))
67	66	ex 402	. . . . . . 7 |- (((K e. HL /\ X e. B) /\ x e. A) -> (xL(U` {y e. A | yLX}) -> -. xL((oc` K)` (U` {y e. A | yLX}))))
68	34, 67	syld 30	. . . . . 6 |- (((K e. HL /\ X e. B) /\ x e. A) -> (xLX -> -. xL((oc` K)` (U` {y e. A | yLX}))))
69		imnan 261	. . . . . 6 |- ((xLX -> -. xL((oc` K)` (U` {y e. A | yLX}))) <-> -. (xLX /\ xL((oc` K)` (U` {y e. A | yLX}))))
70	68, 69	sylib 215	. . . . 5 |- (((K e. HL /\ X e. B) /\ x e. A) -> -. (xLX /\ xL((oc` K)` (U` {y e. A | yLX}))))
71		simplr 449	. . . . . 6 |- (((K e. HL /\ X e. B) /\ x e. A) -> X e. B)
72	4, 10, 52	latlem12 16873	. . . . . 6 |- ((K e. LatNEW /\ (x e. B /\ X e. B /\ ((oc` K)` (U` {y e. A | yLX})) e. B)) -> ((xLX /\ xL((oc` K)` (U` {y e. A | yLX}))) <-> xL(X(meet` K)((oc` K)` (U` {y e. A | yLX})))))
73	42, 44, 71, 51, 72	syl13anc 1102	. . . . 5 |- (((K e. HL /\ X e. B) /\ x e. A) -> ((xLX /\ xL((oc` K)` (U` {y e. A | yLX}))) <-> xL(X(meet` K)((oc` K)` (U` {y e. A | yLX})))))
74	70, 73	mtbid 782	. . . 4 |- (((K e. HL /\ X e. B) /\ x e. A) -> -. xL(X(meet` K)((oc` K)` (U` {y e. A | yLX}))))
75	74	nrexdv 2193	. . 3 |- ((K e. HL /\ X e. B) -> -. E.x e. A xL(X(meet` K)((oc` K)` (U` {y e. A | yLX}))))
76		hlatl 17023	. . . . . . 7 |- (K e. HL -> K e. AtLat)
77	76	ad2antrr 440	. . . . . 6 |- (((K e. HL /\ X e. B) /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K)) -> K e. AtLat)
78	41	adantr 425	. . . . . . . 8 |- ((K e. HL /\ X e. B) -> K e. LatNEW)
79		simpr 350	. . . . . . . 8 |- ((K e. HL /\ X e. B) -> X e. B)
80	50	adantr 425	. . . . . . . 8 |- ((K e. HL /\ X e. B) -> ((oc` K)` (U` {y e. A | yLX})) e. B)
81	4, 52	latmcl 16853	. . . . . . . 8 |- ((K e. LatNEW /\ X e. B /\ ((oc` K)` (U` {y e. A | yLX})) e. B) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) e. B)
82	78, 79, 80, 81	syl111anc 1100	. . . . . . 7 |- ((K e. HL /\ X e. B) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) e. B)
83	82	adantr 425	. . . . . 6 |- (((K e. HL /\ X e. B) /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K)) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) e. B)
84		simpr 350	. . . . . 6 |- (((K e. HL /\ X e. B) /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K)) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K))
85	4, 10, 35, 5	atlatex 17013	. . . . . 6 |- ((K e. AtLat /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) e. B /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K)) -> E.x e. A xL(X(meet` K)((oc` K)` (U` {y e. A | yLX}))))
86	77, 83, 84, 85	syl111anc 1100	. . . . 5 |- (((K e. HL /\ X e. B) /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K)) -> E.x e. A xL(X(meet` K)((oc` K)` (U` {y e. A | yLX}))))
87	86	ex 402	. . . 4 |- ((K e. HL /\ X e. B) -> ((X(meet` K)((oc` K)` (U` {y e. A | yLX}))) =/= (0.` K) -> E.x e. A xL(X(meet` K)((oc` K)` (U` {y e. A | yLX})))))
88	87	necon1bd 2080	. . 3 |- ((K e. HL /\ X e. B) -> (-. E.x e. A xL(X(meet` K)((oc` K)` (U` {y e. A | yLX}))) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K)))
89	75, 88	mpd 29	. 2 |- ((K e. HL /\ X e. B) -> (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K))
90	16	adantr 425	. . 3 |- ((K e. HL /\ X e. B) -> K e. OML)
91	46	adantr 425	. . 3 |- ((K e. HL /\ X e. B) -> (U` {y e. A | yLX}) e. B)
92	4, 10, 52, 48, 35	omllaw3 16966	. . 3 |- ((K e. OML /\ (U` {y e. A | yLX}) e. B /\ X e. B) -> (((U` {y e. A | yLX})LX /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K)) -> (U` {y e. A | yLX}) = X))
93	90, 91, 79, 92	syl111anc 1100	. 2 |- ((K e. HL /\ X e. B) -> (((U` {y e. A | yLX})LX /\ (X(meet` K)((oc` K)` (U` {y e. A | yLX}))) = (0.` K)) -> (U` {y e. A | yLX}) = X))
94	22, 89, 93	mp2and 767	1 |- ((K e. HL /\ X e. B) -> (U` {y e. A | yLX}) = X)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  {crab 2108   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  PosetNEWcpo 16760  lubclub 16764  meetcmee 16767  0.cp0 16832  LatNEWclat 16834  CLatccla 16835  occoc 16836  OPcops 16837  OLcol 16839  OMLcoml 16840  AtomsNEWcatm 16981  AtLatcal 16982  HLchlt 16983

This theorem is referenced by:  hlatle 17048  pmaple 17241  pol1 17323  polpmap 17324  pmaplub 17334

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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