Theorem pmapfval 17236

Description: The projective map of a Hilbert lattice.

Hypotheses

Ref	Expression
pmapfval.b	|- B = (base` K)
pmapfval.l	|- L = (le` K)
pmapfval.a	|- A = (AtomsNEW` K)
pmapfval.m	|- M = (pmap` K)

Assertion

Ref	Expression
pmapfval	|- (K e. C -> M = (x e. B |-> {a e. A | aLx}))

Distinct variable groups:   A,a   x,B   x,a,K

Proof of Theorem pmapfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (K e. C -> K e. _V)
2		fveq2 4681	. . . . . 6 |- (h = K -> (base` h) = (base` K))
3		pmapfval.b	. . . . . 6 |- B = (base` K)
4	2, 3	syl6eqr 1946	. . . . 5 |- (h = K -> (base` h) = B)
5		fveq2 4681	. . . . . . 7 |- (h = K -> (AtomsNEW` h) = (AtomsNEW` K))
6		pmapfval.a	. . . . . . 7 |- A = (AtomsNEW` K)
7	5, 6	syl6eqr 1946	. . . . . 6 |- (h = K -> (AtomsNEW` h) = A)
8		fveq2 4681	. . . . . . . 8 |- (h = K -> (le` h) = (le` K))
9		pmapfval.l	. . . . . . . 8 |- L = (le` K)
10	8, 9	syl6eqr 1946	. . . . . . 7 |- (h = K -> (le` h) = L)
11	10	breqd 3349	. . . . . 6 |- (h = K -> (a(le` h)x <-> aLx))
12	7, 11	rabeqbidv 2290	. . . . 5 |- (h = K -> {a e. (AtomsNEW` h) | a(le` h)x} = {a e. A | aLx})
13	4, 12	mpteq12dv 5008	. . . 4 |- (h = K -> (x e. (base` h) |-> {a e. (AtomsNEW` h) | a(le` h)x}) = (x e. B |-> {a e. A | aLx}))
14		df-pmap 17218	. . . 4 |- pmap = (h e. _V |-> (x e. (base` h) |-> {a e. (AtomsNEW` h) | a(le` h)x}))
15		fvex 4689	. . . . . 6 |- (base` K) e. _V
16	3, 15	eqeltri 1967	. . . . 5 |- B e. _V
17		mptexg 5012	. . . . 5 |- (B e. _V -> (x e. B |-> {a e. A | aLx}) e. _V)
18	16, 17	ax-mp 7	. . . 4 |- (x e. B |-> {a e. A | aLx}) e. _V
19	13, 14, 18	fvmpt 5015	. . 3 |- (K e. _V -> (pmap` K) = (x e. B |-> {a e. A | aLx}))
20		pmapfval.m	. . 3 |- M = (pmap` K)
21	19, 20	syl5eq 1940	. 2 |- (K e. _V -> M = (x e. B |-> {a e. A | aLx}))
22	1, 21	syl 12	1 |- (K e. C -> M = (x e. B |-> {a e. A | aLx}))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {crab 2108  _Vcvv 2292   class class class wbr 3338  ` cfv 3998   e. cmpt 5004  basecbs 16758  lecple 16759  AtomsNEWcatm 16981  pmapcpmap 17214

This theorem is referenced by:  pmapval 17237

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-rex 2110  df-rab 2112  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-fn 4009  df-fv 4014  df-mpt 5006  df-pmap 17218

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