Theorem polfval 17317

Description: The projective subspace polarity function.

Hypotheses

Ref	Expression
polfval.o	|- O = (oc` K)
polfval.a	|- A = (AtomsNEW` K)
polfval.m	|- M = (pmap` K)
polfval.p	|- P = (_|_P` K)

Assertion

Ref	Expression
polfval	|- (K e. B -> P = (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))))

Distinct variable groups:   A,m   m,p,K

Proof of Theorem polfval

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		polfval.a	. . . . . . 7 |- A = (AtomsNEW` K)
4	2, 3	syl6eqr 1946	. . . . . 6 |- (h = K -> (AtomsNEW` h) = A)
5		pweq 3036	. . . . . 6 |- ((AtomsNEW` h) = A -> ~P(AtomsNEW` h) = ~PA)
6	4, 5	syl 12	. . . . 5 |- (h = K -> ~P(AtomsNEW` h) = ~PA)
7		fveq2 4681	. . . . . . . . . 10 |- (h = K -> (pmap` h) = (pmap` K))
8		polfval.m	. . . . . . . . . 10 |- M = (pmap` K)
9	7, 8	syl6eqr 1946	. . . . . . . . 9 |- (h = K -> (pmap` h) = M)
10		fveq2 4681	. . . . . . . . . . 11 |- (h = K -> (oc` h) = (oc` K))
11		polfval.o	. . . . . . . . . . 11 |- O = (oc` K)
12	10, 11	syl6eqr 1946	. . . . . . . . . 10 |- (h = K -> (oc` h) = O)
13	12	fveq1d 4683	. . . . . . . . 9 |- (h = K -> ((oc` h)` p) = (O` p))
14	9, 13	fveq12d 10152	. . . . . . . 8 |- (h = K -> ((pmap` h)` ((oc` h)` p)) = (M` (O` p)))
15	14	adantr 425	. . . . . . 7 |- ((h = K /\ p e. m) -> ((pmap` h)` ((oc` h)` p)) = (M` (O` p)))
16	15	iineq2dv 3280	. . . . . 6 |- (h = K -> |^|_p e. m ((pmap` h)` ((oc` h)` p)) = |^|_p e. m (M` (O` p)))
17	4, 16	ineq12d 2797	. . . . 5 |- (h = K -> ((AtomsNEW` h) i^i |^|_p e. m ((pmap` h)` ((oc` h)` p))) = (A i^i |^|_p e. m (M` (O` p))))
18	6, 17	mpteq12dv 5008	. . . 4 |- (h = K -> (m e. ~P(AtomsNEW` h) |-> ((AtomsNEW` h) i^i |^|_p e. m ((pmap` h)` ((oc` h)` p)))) = (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))))
19		df-polarity 17316	. . . 4 |- _|_P = (h e. _V |-> (m e. ~P(AtomsNEW` h) |-> ((AtomsNEW` h) i^i |^|_p e. m ((pmap` h)` ((oc` h)` p)))))
20		fvex 4689	. . . . . . 7 |- (AtomsNEW` K) e. _V
21	3, 20	eqeltri 1967	. . . . . 6 |- A e. _V
22	21	pwex 3487	. . . . 5 |- ~PA e. _V
23		mptexg 5012	. . . . 5 |- (~PA e. _V -> (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))) e. _V)
24	22, 23	ax-mp 7	. . . 4 |- (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))) e. _V
25	18, 19, 24	fvmpt 5015	. . 3 |- (K e. _V -> (_|_P` K) = (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))))
26		polfval.p	. . 3 |- P = (_|_P` K)
27	25, 26	syl5eq 1940	. 2 |- (K e. _V -> P = (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))))
28	1, 27	syl 12	1 |- (K e. B -> P = (m e. ~PA |-> (A i^i |^|_p e. m (M` (O` p)))))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592  ~Pcpw 3032  |^|_ciin 3256  ` cfv 3998   e. cmpt 5004  occoc 16836  AtomsNEWcatm 16981  pmapcpmap 17214  _|_Pcpol 17315

This theorem is referenced by:  polval 17318

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-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-iin 3258  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-polarity 17316

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