Theorem pmapsub 17250

Description: The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62.

Hypotheses

Ref	Expression
pmapsub.b	|- B = (base` K)
pmapsub.s	|- S = (PSubSp` K)
pmapsub.m	|- M = (pmap` K)

Assertion

Ref	Expression
pmapsub	|- ((K e. LatNEW /\ X e. B) -> (M` X) e. S)

Proof of Theorem pmapsub

Step	Hyp	Ref	Expression
1		pmapsub.b	. . 3 |- B = (base` K)
2		eqid 1884	. . 3 |- (le` K) = (le` K)
3		eqid 1884	. . 3 |- (AtomsNEW` K) = (AtomsNEW` K)
4		pmapsub.m	. . 3 |- M = (pmap` K)
5	1, 2, 3, 4	pmapval 17237	. 2 |- ((K e. LatNEW /\ X e. B) -> (M` X) = {c e. (AtomsNEW` K) | c(le` K)X})
6		eqid 1884	. . . . . . . . . . . . . . . . 17 |- (join` K) = (join` K)
7	1, 2, 6	latjle12 16863	. . . . . . . . . . . . . . . 16 |- ((K e. LatNEW /\ (p e. B /\ q e. B /\ X e. B)) -> ((p(le` K)X /\ q(le` K)X) <-> (p(join` K)q)(le` K)X))
8	7	biimpd 170	. . . . . . . . . . . . . . 15 |- ((K e. LatNEW /\ (p e. B /\ q e. B /\ X e. B)) -> ((p(le` K)X /\ q(le` K)X) -> (p(join` K)q)(le` K)X))
9	8	3exp2 1086	. . . . . . . . . . . . . 14 |- (K e. LatNEW -> (p e. B -> (q e. B -> (X e. B -> ((p(le` K)X /\ q(le` K)X) -> (p(join` K)q)(le` K)X)))))
10	9	imp3a 388	. . . . . . . . . . . . 13 |- (K e. LatNEW -> ((p e. B /\ q e. B) -> (X e. B -> ((p(le` K)X /\ q(le` K)X) -> (p(join` K)q)(le` K)X))))
11	10	com23 36	. . . . . . . . . . . 12 |- (K e. LatNEW -> (X e. B -> ((p e. B /\ q e. B) -> ((p(le` K)X /\ q(le` K)X) -> (p(join` K)q)(le` K)X))))
12	11	imp43 397	. . . . . . . . . . 11 |- (((K e. LatNEW /\ X e. B) /\ ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X))) -> (p(join` K)q)(le` K)X)
13	12	adantr 425	. . . . . . . . . 10 |- ((((K e. LatNEW /\ X e. B) /\ ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X))) /\ r e. B) -> (p(join` K)q)(le` K)X)
14	1, 2	postrNEW 16777	. . . . . . . . . . . . . . . 16 |- ((K e. PosetNEW /\ (r e. B /\ (p(join` K)q) e. B /\ X e. B)) -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X))
15		latpos 16851	. . . . . . . . . . . . . . . 16 |- (K e. LatNEW -> K e. PosetNEW)
16	14, 15	sylan 497	. . . . . . . . . . . . . . 15 |- ((K e. LatNEW /\ (r e. B /\ (p(join` K)q) e. B /\ X e. B)) -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X))
17	16	3exp2 1086	. . . . . . . . . . . . . 14 |- (K e. LatNEW -> (r e. B -> ((p(join` K)q) e. B -> (X e. B -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X)))))
18	17	com24 41	. . . . . . . . . . . . 13 |- (K e. LatNEW -> (X e. B -> ((p(join` K)q) e. B -> (r e. B -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X)))))
19	1, 6	latjcl 16852	. . . . . . . . . . . . . 14 |- ((K e. LatNEW /\ p e. B /\ q e. B) -> (p(join` K)q) e. B)
20	19	3expib 1070	. . . . . . . . . . . . 13 |- (K e. LatNEW -> ((p e. B /\ q e. B) -> (p(join` K)q) e. B))
21	18, 20	syl5d 66	. . . . . . . . . . . 12 |- (K e. LatNEW -> (X e. B -> ((p e. B /\ q e. B) -> (r e. B -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X)))))
22	21	imp41 395	. . . . . . . . . . 11 |- ((((K e. LatNEW /\ X e. B) /\ (p e. B /\ q e. B)) /\ r e. B) -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X))
23	22	adantlrr 435	. . . . . . . . . 10 |- ((((K e. LatNEW /\ X e. B) /\ ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X))) /\ r e. B) -> ((r(le` K)(p(join` K)q) /\ (p(join` K)q)(le` K)X) -> r(le` K)X))
24	13, 23	mpan2d 766	. . . . . . . . 9 |- ((((K e. LatNEW /\ X e. B) /\ ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X))) /\ r e. B) -> (r(le` K)(p(join` K)q) -> r(le` K)X))
25		breq1 3341	. . . . . . . . . . . . . 14 |- (c = p -> (c(le` K)X <-> p(le` K)X))
26	25	elrab 2414	. . . . . . . . . . . . 13 |- (p e. {c e. (AtomsNEW` K) | c(le` K)X} <-> (p e. (AtomsNEW` K) /\ p(le` K)X))
27	1, 3	atombase 17003	. . . . . . . . . . . . . 14 |- (p e. (AtomsNEW` K) -> p e. B)
28	27	anim1i 361	. . . . . . . . . . . . 13 |- ((p e. (AtomsNEW` K) /\ p(le` K)X) -> (p e. B /\ p(le` K)X))
29	26, 28	sylbi 216	. . . . . . . . . . . 12 |- (p e. {c e. (AtomsNEW` K) | c(le` K)X} -> (p e. B /\ p(le` K)X))
30		breq1 3341	. . . . . . . . . . . . . 14 |- (c = q -> (c(le` K)X <-> q(le` K)X))
31	30	elrab 2414	. . . . . . . . . . . . 13 |- (q e. {c e. (AtomsNEW` K) | c(le` K)X} <-> (q e. (AtomsNEW` K) /\ q(le` K)X))
32	1, 3	atombase 17003	. . . . . . . . . . . . . 14 |- (q e. (AtomsNEW` K) -> q e. B)
33	32	anim1i 361	. . . . . . . . . . . . 13 |- ((q e. (AtomsNEW` K) /\ q(le` K)X) -> (q e. B /\ q(le` K)X))
34	31, 33	sylbi 216	. . . . . . . . . . . 12 |- (q e. {c e. (AtomsNEW` K) | c(le` K)X} -> (q e. B /\ q(le` K)X))
35	29, 34	anim12i 360	. . . . . . . . . . 11 |- ((p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X}) -> ((p e. B /\ p(le` K)X) /\ (q e. B /\ q(le` K)X)))
36		an4 564	. . . . . . . . . . 11 |- (((p e. B /\ p(le` K)X) /\ (q e. B /\ q(le` K)X)) <-> ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X)))
37	35, 36	sylib 215	. . . . . . . . . 10 |- ((p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X}) -> ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X)))
38	37	anim2i 362	. . . . . . . . 9 |- (((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) -> ((K e. LatNEW /\ X e. B) /\ ((p e. B /\ q e. B) /\ (p(le` K)X /\ q(le` K)X))))
39	1, 3	atombase 17003	. . . . . . . . 9 |- (r e. (AtomsNEW` K) -> r e. B)
40	24, 38, 39	syl2an 503	. . . . . . . 8 |- ((((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) /\ r e. (AtomsNEW` K)) -> (r(le` K)(p(join` K)q) -> r(le` K)X))
41		simpr 350	. . . . . . . 8 |- ((((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) /\ r e. (AtomsNEW` K)) -> r e. (AtomsNEW` K))
42	40, 41	jctild 662	. . . . . . 7 |- ((((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) /\ r e. (AtomsNEW` K)) -> (r(le` K)(p(join` K)q) -> (r e. (AtomsNEW` K) /\ r(le` K)X)))
43		breq1 3341	. . . . . . . 8 |- (c = r -> (c(le` K)X <-> r(le` K)X))
44	43	elrab 2414	. . . . . . 7 |- (r e. {c e. (AtomsNEW` K) | c(le` K)X} <-> (r e. (AtomsNEW` K) /\ r(le` K)X))
45	42, 44	syl6ibr 230	. . . . . 6 |- ((((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) /\ r e. (AtomsNEW` K)) -> (r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X}))
46	45	r19.21aiva 2176	. . . . 5 |- (((K e. LatNEW /\ X e. B) /\ (p e. {c e. (AtomsNEW` K) | c(le` K)X} /\ q e. {c e. (AtomsNEW` K) | c(le` K)X})) -> A.r e. (AtomsNEW` K)(r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X}))
47	46	r19.21aivva 15653	. . . 4 |- ((K e. LatNEW /\ X e. B) -> A.p e. {c e. (AtomsNEW` K) | c(le` K)X}A.q e. {c e. (AtomsNEW` K) | c(le` K)X}A.r e. (AtomsNEW` K)(r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X}))
48		ssrab2 2692	. . . 4 |- {c e. (AtomsNEW` K) | c(le` K)X} C_ (AtomsNEW` K)
49	47, 48	jctil 316	. . 3 |- ((K e. LatNEW /\ X e. B) -> ({c e. (AtomsNEW` K) | c(le` K)X} C_ (AtomsNEW` K) /\ A.p e. {c e. (AtomsNEW` K) | c(le` K)X}A.q e. {c e. (AtomsNEW` K) | c(le` K)X}A.r e. (AtomsNEW` K)(r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X})))
50		pmapsub.s	. . . . 5 |- S = (PSubSp` K)
51	2, 6, 3, 50	ispsubsp 17226	. . . 4 |- (K e. LatNEW -> ({c e. (AtomsNEW` K) | c(le` K)X} e. S <-> ({c e. (AtomsNEW` K) | c(le` K)X} C_ (AtomsNEW` K) /\ A.p e. {c e. (AtomsNEW` K) | c(le` K)X}A.q e. {c e. (AtomsNEW` K) | c(le` K)X}A.r e. (AtomsNEW` K)(r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X}))))
52	51	adantr 425	. . 3 |- ((K e. LatNEW /\ X e. B) -> ({c e. (AtomsNEW` K) | c(le` K)X} e. S <-> ({c e. (AtomsNEW` K) | c(le` K)X} C_ (AtomsNEW` K) /\ A.p e. {c e. (AtomsNEW` K) | c(le` K)X}A.q e. {c e. (AtomsNEW` K) | c(le` K)X}A.r e. (AtomsNEW` K)(r(le` K)(p(join` K)q) -> r e. {c e. (AtomsNEW` K) | c(le` K)X}))))
53	49, 52	mpbird 213	. 2 |- ((K e. LatNEW /\ X e. B) -> {c e. (AtomsNEW` K) | c(le` K)X} e. S)
54	5, 53	eqeltrd 1971	1 |- ((K e. LatNEW /\ X e. B) -> (M` X) e. S)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  basecbs 16758  lecple 16759  PosetNEWcpo 16760  joincjn 16766  LatNEWclat 16834  AtomsNEWcatm 16981  PSubSpcpsubsp 17213  pmapcpmap 17214

This theorem is referenced by:  polsub 17320  pl42lem4 17410

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-lub 16799  df-join 16801  df-lat 16847  df-atoms 16985  df-psubsp 17217  df-pmap 17218

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