Theorem ocvfval 17206

Description: The orthocomplement operation.

Hypotheses

Ref	Expression
ocvfval.v	|- V = (vbase` H)
ocvfval.i	|- I = (ip` H)
ocvfval.z	|- Z = (0vNEW` H)
ocvfval.o	|- O = (ocv` H)

Assertion

Ref	Expression
ocvfval	|- (H e. A -> O = (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}))

Distinct variable groups:   x,s,y,H   V,s,x

Proof of Theorem ocvfval

Step	Hyp	Ref	Expression
1		elisset 2299	. 2 |- (H e. A -> H e. _V)
2		fveq2 4681	. . . . . . 7 |- (h = H -> (vbase` h) = (vbase` H))
3		ocvfval.v	. . . . . . 7 |- V = (vbase` H)
4	2, 3	syl6eqr 1946	. . . . . 6 |- (h = H -> (vbase` h) = V)
5		pweq 3036	. . . . . 6 |- ((vbase` h) = V -> ~P(vbase` h) = ~PV)
6	4, 5	syl 12	. . . . 5 |- (h = H -> ~P(vbase` h) = ~PV)
7		fveq2 4681	. . . . . . . . . 10 |- (h = H -> (ip` h) = (ip` H))
8		ocvfval.i	. . . . . . . . . 10 |- I = (ip` H)
9	7, 8	syl6eqr 1946	. . . . . . . . 9 |- (h = H -> (ip` h) = I)
10	9	opreqd 4899	. . . . . . . 8 |- (h = H -> (x(ip` h)y) = (xIy))
11		fveq2 4681	. . . . . . . . 9 |- (h = H -> (0vNEW` h) = (0vNEW` H))
12		ocvfval.z	. . . . . . . . 9 |- Z = (0vNEW` H)
13	11, 12	syl6eqr 1946	. . . . . . . 8 |- (h = H -> (0vNEW` h) = Z)
14	10, 13	eqeq12d 1899	. . . . . . 7 |- (h = H -> ((x(ip` h)y) = (0vNEW` h) <-> (xIy) = Z))
15	14	ralbidv 2123	. . . . . 6 |- (h = H -> (A.y e. s (x(ip` h)y) = (0vNEW` h) <-> A.y e. s (xIy) = Z))
16	4, 15	rabeqbidv 2290	. . . . 5 |- (h = H -> {x e. (vbase` h) | A.y e. s (x(ip` h)y) = (0vNEW` h)} = {x e. V | A.y e. s (xIy) = Z})
17	6, 16	mpteq12dv 5008	. . . 4 |- (h = H -> (s e. ~P(vbase` h) |-> {x e. (vbase` h) | A.y e. s (x(ip` h)y) = (0vNEW` h)}) = (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}))
18		df-ocv 17201	. . . 4 |- ocv = (h e. _V |-> (s e. ~P(vbase` h) |-> {x e. (vbase` h) | A.y e. s (x(ip` h)y) = (0vNEW` h)}))
19		fvex 4689	. . . . . . 7 |- (vbase` H) e. _V
20	3, 19	eqeltri 1967	. . . . . 6 |- V e. _V
21	20	pwex 3487	. . . . 5 |- ~PV e. _V
22		mptexg 5012	. . . . 5 |- (~PV e. _V -> (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}) e. _V)
23	21, 22	ax-mp 7	. . . 4 |- (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}) e. _V
24	17, 18, 23	fvmpt 5015	. . 3 |- (H e. _V -> (ocv` H) = (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}))
25		ocvfval.o	. . 3 |- O = (ocv` H)
26	24, 25	syl5eq 1940	. 2 |- (H e. _V -> O = (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}))
27	1, 26	syl 12	1 |- (H e. A -> O = (s e. ~PV |-> {x e. V | A.y e. s (xIy) = Z}))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  _Vcvv 2292  ~Pcpw 3032  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  vbasecvbase 17180  0vNEWczv 17189  ipcipr 17191  ocvcocv 17197

This theorem is referenced by:  ocvval 17207

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-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-opr 4886  df-mpt 5006  df-ocv 17201

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