Theorem lineset 17219

Description: The set of lines in a Hilbert lattice.

Hypotheses

Ref	Expression
lineset.l	|- L = (le` K)
lineset.j	|- J = (join` K)
lineset.a	|- A = (AtomsNEW` K)
lineset.n	|- N = (Lines` K)

Assertion

Ref	Expression
lineset	|- (K e. B -> N = {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})})

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

Proof of Theorem lineset

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. B -> K e. _V)
2		fveq2 4681	. . . . . . 7 |- (h = K -> (AtomsNEW` h) = (AtomsNEW` K))
3		lineset.a	. . . . . . 7 |- A = (AtomsNEW` K)
4	2, 3	syl6eqr 1946	. . . . . 6 |- (h = K -> (AtomsNEW` h) = A)
5		fveq2 4681	. . . . . . . . . . . . 13 |- (h = K -> (le` h) = (le` K))
6		lineset.l	. . . . . . . . . . . . 13 |- L = (le` K)
7	5, 6	syl6eqr 1946	. . . . . . . . . . . 12 |- (h = K -> (le` h) = L)
8	7	breqd 3349	. . . . . . . . . . 11 |- (h = K -> (p(le` h)(q(join` h)r) <-> pL(q(join` h)r)))
9		fveq2 4681	. . . . . . . . . . . . . 14 |- (h = K -> (join` h) = (join` K))
10		lineset.j	. . . . . . . . . . . . . 14 |- J = (join` K)
11	9, 10	syl6eqr 1946	. . . . . . . . . . . . 13 |- (h = K -> (join` h) = J)
12	11	opreqd 4899	. . . . . . . . . . . 12 |- (h = K -> (q(join` h)r) = (qJr))
13	12	breq2d 3350	. . . . . . . . . . 11 |- (h = K -> (pL(q(join` h)r) <-> pL(qJr)))
14	8, 13	bitrd 587	. . . . . . . . . 10 |- (h = K -> (p(le` h)(q(join` h)r) <-> pL(qJr)))
15	4, 14	rabeqbidv 2290	. . . . . . . . 9 |- (h = K -> {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)} = {p e. A | pL(qJr)})
16	15	eqeq2d 1895	. . . . . . . 8 |- (h = K -> (s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)} <-> s = {p e. A | pL(qJr)}))
17	16	anbi2d 678	. . . . . . 7 |- (h = K -> ((q =/= r /\ s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)}) <-> (q =/= r /\ s = {p e. A | pL(qJr)})))
18	4, 17	rexeqbidv 2275	. . . . . 6 |- (h = K -> (E.r e. (AtomsNEW` h)(q =/= r /\ s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)}) <-> E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})))
19	4, 18	rexeqbidv 2275	. . . . 5 |- (h = K -> (E.q e. (AtomsNEW` h)E.r e. (AtomsNEW` h)(q =/= r /\ s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)}) <-> E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})))
20	19	abbidv 2008	. . . 4 |- (h = K -> {s | E.q e. (AtomsNEW` h)E.r e. (AtomsNEW` h)(q =/= r /\ s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)})} = {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})})
21		df-lines 17215	. . . 4 |- Lines = (h e. _V |-> {s | E.q e. (AtomsNEW` h)E.r e. (AtomsNEW` h)(q =/= r /\ s = {p e. (AtomsNEW` h) | p(le` h)(q(join` h)r)})})
22		fvex 4689	. . . . . 6 |- (AtomsNEW` K) e. _V
23	3, 22	eqeltri 1967	. . . . 5 |- A e. _V
24		df-sn 3049	. . . . . . 7 |- {{p e. A | pL(qJr)}} = {s | s = {p e. A | pL(qJr)}}
25		snex 3492	. . . . . . 7 |- {{p e. A | pL(qJr)}} e. _V
26	24, 25	eqeltrri 1968	. . . . . 6 |- {s | s = {p e. A | pL(qJr)}} e. _V
27		simpr 350	. . . . . . 7 |- ((q =/= r /\ s = {p e. A | pL(qJr)}) -> s = {p e. A | pL(qJr)})
28	27	ss2abi 2679	. . . . . 6 |- {s | (q =/= r /\ s = {p e. A | pL(qJr)})} C_ {s | s = {p e. A | pL(qJr)}}
29	26, 28	ssexi 3456	. . . . 5 |- {s | (q =/= r /\ s = {p e. A | pL(qJr)})} e. _V
30	23, 23, 29	ab2rexex2 4972	. . . 4 |- {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})} e. _V
31	20, 21, 30	fvmpt 5015	. . 3 |- (K e. _V -> (Lines` K) = {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})})
32		lineset.n	. . 3 |- N = (Lines` K)
33	31, 32	syl5eq 1940	. 2 |- (K e. _V -> N = {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})})
34	1, 33	syl 12	1 |- (K e. B -> N = {s | E.q e. A E.r e. A (q =/= r /\ s = {p e. A | pL(qJr)})})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106  {crab 2108  _Vcvv 2292  {csn 3044   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  lecple 16759  joincjn 16766  AtomsNEWcatm 16981  Linesclines 17211

This theorem is referenced by:  isline 17220

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-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-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-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  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-lines 17215

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