Theorem psubspset 17225

Description: The set of projective subspaces in a Hilbert lattice.

Hypotheses

Ref	Expression
psubspset.l	|- L = (le` K)
psubspset.j	|- J = (join` K)
psubspset.a	|- A = (AtomsNEW` K)
psubspset.s	|- S = (PSubSp` K)

Assertion

Ref	Expression
psubspset	|- (K e. B -> S = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))})

Distinct variable groups:   s,r,A   q,p,r,s,K

Proof of Theorem psubspset

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. B -> K e. _V)
2		fveq2 4681	. . . . . . . 8 |- (h = K -> (AtomsNEW` h) = (AtomsNEW` K))
3		psubspset.a	. . . . . . . 8 |- A = (AtomsNEW` K)
4	2, 3	syl6eqr 1946	. . . . . . 7 |- (h = K -> (AtomsNEW` h) = A)
5	4	sseq2d 2645	. . . . . 6 |- (h = K -> (s C_ (AtomsNEW` h) <-> s C_ A))
6		fveq2 4681	. . . . . . . . . . . . 13 |- (h = K -> (join` h) = (join` K))
7		psubspset.j	. . . . . . . . . . . . 13 |- J = (join` K)
8	6, 7	syl6eqr 1946	. . . . . . . . . . . 12 |- (h = K -> (join` h) = J)
9	8	opreqd 4899	. . . . . . . . . . 11 |- (h = K -> (p(join` h)q) = (pJq))
10	9	breq2d 3350	. . . . . . . . . 10 |- (h = K -> (r(le` h)(p(join` h)q) <-> r(le` h)(pJq)))
11		fveq2 4681	. . . . . . . . . . . 12 |- (h = K -> (le` h) = (le` K))
12		psubspset.l	. . . . . . . . . . . 12 |- L = (le` K)
13	11, 12	syl6eqr 1946	. . . . . . . . . . 11 |- (h = K -> (le` h) = L)
14	13	breqd 3349	. . . . . . . . . 10 |- (h = K -> (r(le` h)(pJq) <-> rL(pJq)))
15	10, 14	bitrd 587	. . . . . . . . 9 |- (h = K -> (r(le` h)(p(join` h)q) <-> rL(pJq)))
16	15	imbi1d 675	. . . . . . . 8 |- (h = K -> ((r(le` h)(p(join` h)q) -> r e. s) <-> (rL(pJq) -> r e. s)))
17	4, 16	raleqbidv 2274	. . . . . . 7 |- (h = K -> (A.r e. (AtomsNEW` h)(r(le` h)(p(join` h)q) -> r e. s) <-> A.r e. A (rL(pJq) -> r e. s)))
18	17	2ralbidv 2140	. . . . . 6 |- (h = K -> (A.p e. s A.q e. s A.r e. (AtomsNEW` h)(r(le` h)(p(join` h)q) -> r e. s) <-> A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s)))
19	5, 18	anbi12d 690	. . . . 5 |- (h = K -> ((s C_ (AtomsNEW` h) /\ A.p e. s A.q e. s A.r e. (AtomsNEW` h)(r(le` h)(p(join` h)q) -> r e. s)) <-> (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))))
20	19	abbidv 2008	. . . 4 |- (h = K -> {s | (s C_ (AtomsNEW` h) /\ A.p e. s A.q e. s A.r e. (AtomsNEW` h)(r(le` h)(p(join` h)q) -> r e. s))} = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))})
21		df-psubsp 17217	. . . 4 |- PSubSp = (h e. _V |-> {s | (s C_ (AtomsNEW` h) /\ A.p e. s A.q e. s A.r e. (AtomsNEW` h)(r(le` h)(p(join` h)q) -> r e. s))})
22		fvex 4689	. . . . . . 7 |- (AtomsNEW` K) e. _V
23	3, 22	eqeltri 1967	. . . . . 6 |- A e. _V
24	23	pwex 3487	. . . . 5 |- ~PA e. _V
25		df-pw 3035	. . . . . . . . 9 |- ~PA = {s | s C_ A}
26	25	abeq2i 2001	. . . . . . . 8 |- (s e. ~PA <-> s C_ A)
27	26	anbi1i 539	. . . . . . 7 |- ((s e. ~PA /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s)) <-> (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s)))
28	27	abbii 2006	. . . . . 6 |- {s | (s e. ~PA /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))} = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))}
29		ssab2 2691	. . . . . 6 |- {s | (s e. ~PA /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))} C_ ~PA
30	28, 29	eqsstr3i 2648	. . . . 5 |- {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))} C_ ~PA
31	24, 30	ssexi 3456	. . . 4 |- {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))} e. _V
32	20, 21, 31	fvmpt 5015	. . 3 |- (K e. _V -> (PSubSp` K) = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))})
33		psubspset.s	. . 3 |- S = (PSubSp` K)
34	32, 33	syl5eq 1940	. 2 |- (K e. _V -> S = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))})
35	1, 34	syl 12	1 |- (K e. B -> S = {s | (s C_ A /\ A.p e. s A.q e. s A.r e. A (rL(pJq) -> r e. s))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  lecple 16759  joincjn 16766  AtomsNEWcatm 16981  PSubSpcpsubsp 17213

This theorem is referenced by:  ispsubsp 17226

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-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-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-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  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-fv 4014  df-opr 4886  df-mpt 5006  df-psubsp 17217

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