Theorem pmaple 17241

Description: The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62.

Hypotheses

Ref	Expression
pmaple.b	|- B = (base` K)
pmaple.l	|- L = (le` K)
pmaple.m	|- M = (pmap` K)

Assertion

Ref	Expression
pmaple	|- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY <-> (M` X) C_ (M` Y)))

Proof of Theorem pmaple

Step	Hyp	Ref	Expression
1		pmaple.b	. . . . . . . . . . . . . . 15 |- B = (base` K)
2		pmaple.l	. . . . . . . . . . . . . . 15 |- L = (le` K)
3	1, 2	postrNEW 16777	. . . . . . . . . . . . . 14 |- ((K e. PosetNEW /\ (p e. B /\ X e. B /\ Y e. B)) -> ((pLX /\ XLY) -> pLY))
4	3	exp4b 410	. . . . . . . . . . . . 13 |- (K e. PosetNEW -> ((p e. B /\ X e. B /\ Y e. B) -> (pLX -> (XLY -> pLY))))
5	4	3expd 1085	. . . . . . . . . . . 12 |- (K e. PosetNEW -> (p e. B -> (X e. B -> (Y e. B -> (pLX -> (XLY -> pLY))))))
6	5	com23 36	. . . . . . . . . . 11 |- (K e. PosetNEW -> (X e. B -> (p e. B -> (Y e. B -> (pLX -> (XLY -> pLY))))))
7	6	com34 40	. . . . . . . . . 10 |- (K e. PosetNEW -> (X e. B -> (Y e. B -> (p e. B -> (pLX -> (XLY -> pLY))))))
8	7	3imp 1061	. . . . . . . . 9 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (p e. B -> (pLX -> (XLY -> pLY))))
9		eqid 1884	. . . . . . . . . 10 |- (AtomsNEW` K) = (AtomsNEW` K)
10	1, 9	atombase 17003	. . . . . . . . 9 |- (p e. (AtomsNEW` K) -> p e. B)
11	8, 10	syl5 20	. . . . . . . 8 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (p e. (AtomsNEW` K) -> (pLX -> (XLY -> pLY))))
12	11	com34 40	. . . . . . 7 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (p e. (AtomsNEW` K) -> (XLY -> (pLX -> pLY))))
13	12	com23 36	. . . . . 6 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (XLY -> (p e. (AtomsNEW` K) -> (pLX -> pLY))))
14	13	r19.21adv 2181	. . . . 5 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (XLY -> A.p e. (AtomsNEW` K)(pLX -> pLY)))
15		hlpos 17027	. . . . 5 |- (K e. HL -> K e. PosetNEW)
16	14, 15	syl3an1 1130	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY -> A.p e. (AtomsNEW` K)(pLX -> pLY)))
17		ss2rab 2683	. . . 4 |- ({p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY} <-> A.p e. (AtomsNEW` K)(pLX -> pLY))
18	16, 17	syl6ibr 230	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY -> {p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY}))
19		hlclat 17022	. . . . . 6 |- (K e. HL -> K e. CLat)
20		ssrab2 2692	. . . . . . . . 9 |- {p e. (AtomsNEW` K) | pLY} C_ (AtomsNEW` K)
21	1, 9	atomssbase 17004	. . . . . . . . 9 |- (AtomsNEW` K) C_ B
22	20, 21	sstri 2626	. . . . . . . 8 |- {p e. (AtomsNEW` K) | pLY} C_ B
23		eqid 1884	. . . . . . . . 9 |- (lub` K) = (lub` K)
24	1, 2, 23	lubss 16897	. . . . . . . 8 |- ((K e. CLat /\ {p e. (AtomsNEW` K) | pLY} C_ B /\ {p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY}) -> ((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY}))
25	22, 24	mp3an2 1179	. . . . . . 7 |- ((K e. CLat /\ {p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY}) -> ((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY}))
26	25	ex 402	. . . . . 6 |- (K e. CLat -> ({p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY} -> ((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY})))
27	19, 26	syl 12	. . . . 5 |- (K e. HL -> ({p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY} -> ((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY})))
28	27	3ad2ant1 897	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> ({p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY} -> ((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY})))
29	1, 2, 23, 9	hlatmstc 17047	. . . . . 6 |- ((K e. HL /\ X e. B) -> ((lub` K)` {p e. (AtomsNEW` K) | pLX}) = X)
30	29	3adant3 896	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> ((lub` K)` {p e. (AtomsNEW` K) | pLX}) = X)
31	1, 2, 23, 9	hlatmstc 17047	. . . . . 6 |- ((K e. HL /\ Y e. B) -> ((lub` K)` {p e. (AtomsNEW` K) | pLY}) = Y)
32	31	3adant2 895	. . . . 5 |- ((K e. HL /\ X e. B /\ Y e. B) -> ((lub` K)` {p e. (AtomsNEW` K) | pLY}) = Y)
33	30, 32	breq12d 3351	. . . 4 |- ((K e. HL /\ X e. B /\ Y e. B) -> (((lub` K)` {p e. (AtomsNEW` K) | pLX})L((lub` K)` {p e. (AtomsNEW` K) | pLY}) <-> XLY))
34	28, 33	sylibd 219	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> ({p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY} -> XLY))
35	18, 34	impbid 574	. 2 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY <-> {p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY}))
36		pmaple.m	. . . . 5 |- M = (pmap` K)
37	1, 2, 9, 36	pmapval 17237	. . . 4 |- ((K e. HL /\ X e. B) -> (M` X) = {p e. (AtomsNEW` K) | pLX})
38	37	3adant3 896	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (M` X) = {p e. (AtomsNEW` K) | pLX})
39	1, 2, 9, 36	pmapval 17237	. . . 4 |- ((K e. HL /\ Y e. B) -> (M` Y) = {p e. (AtomsNEW` K) | pLY})
40	39	3adant2 895	. . 3 |- ((K e. HL /\ X e. B /\ Y e. B) -> (M` Y) = {p e. (AtomsNEW` K) | pLY})
41	38, 40	sseq12d 2646	. 2 |- ((K e. HL /\ X e. B /\ Y e. B) -> ((M` X) C_ (M` Y) <-> {p e. (AtomsNEW` K) | pLX} C_ {p e. (AtomsNEW` K) | pLY}))
42	35, 41	bitr4d 590	1 |- ((K e. HL /\ X e. B /\ Y e. B) -> (XLY <-> (M` X) C_ (M` Y)))

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593   class class class wbr 3338  ` cfv 3998  basecbs 16758  lecple 16759  PosetNEWcpo 16760  lubclub 16764  CLatccla 16835  AtomsNEWcatm 16981  HLchlt 16983  pmapcpmap 17214

This theorem is referenced by:  pmap11 17242  paddun 17337  pmapojoin 17376  pl42 17411

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  df-pmap 17218

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