Theorem ocvfval 21575

Description: The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)

Hypotheses

Ref	Expression
ocvfval.v	⊢ 𝑉 = (Base‘𝑊)
ocvfval.i	⊢ , = (·𝑖‘𝑊)
ocvfval.f	⊢ 𝐹 = (Scalar‘𝑊)
ocvfval.z	⊢ 0 = (0g‘𝐹)
ocvfval.o	⊢ ⊥ = (ocv‘𝑊)

Assertion

Ref	Expression
ocvfval	⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

Distinct variable groups:   𝑥,𝑠,𝑦, 0   𝑉,𝑠,𝑥,𝑦   𝑊,𝑠,𝑥,𝑦   , ,𝑠,𝑥,𝑦

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑠)   ⊥ (𝑥,𝑦,𝑠)   𝑋(𝑥,𝑦,𝑠)

Proof of Theorem ocvfval

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ocvfval.o	. 2 ⊢ ⊥ = (ocv‘𝑊)
2		elex 3468	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6858	. . . . . . 7 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		ocvfval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2782	. . . . . 6 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4580	. . . . 5 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6858	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		ocvfval.i	. . . . . . . . . 10 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2782	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7404	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6858	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		ocvfval.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2782	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6862	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
15		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
16	14, 15	eqtr4di 2782	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
17	10, 16	eqeq12d 2745	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ (𝑥 , 𝑦) = 0 ))
18	17	ralbidv 3156	. . . . . 6 ⊢ (ℎ = 𝑊 → (∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ))
19	5, 18	rabeqbidv 3424	. . . . 5 ⊢ (ℎ = 𝑊 → {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))} = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
20	6, 19	mpteq12dv 5194	. . . 4 ⊢ (ℎ = 𝑊 → (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
21		df-ocv 21572	. . . 4 ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
22		eqid 2729	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
23	4	fvexi 6872	. . . . . . . 8 ⊢ 𝑉 ∈ V
24		ssrab2 4043	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
25	23, 24	elpwi2 5290	. . . . . . 7 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉
26	25	a1i 11	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 → {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ∈ 𝒫 𝑉)
27	22, 26	fmpti 7084	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
28	23	pwex 5335	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
29		fex2 7912	. . . . 5 ⊢ (((𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉 ∧ 𝒫 𝑉 ∈ V ∧ 𝒫 𝑉 ∈ V) → (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) ∈ V)
30	27, 28, 28, 29	mp3an 1463	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
31	20, 21, 30	fvmpt 6968	. . 3 ⊢ (𝑊 ∈ V → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
32	2, 31	syl 17	. 2 ⊢ (𝑊 ∈ 𝑋 → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
33	1, 32	eqtrid 2776	1 ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447  𝒫 cpw 4563   ↦ cmpt 5188  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Scalarcsca 17223  ·_𝑖cip 17225  0_gc0g 17402  ocvcocv 21569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-ocv 21572

This theorem is referenced by:  ocvval  21576  elocv  21577