Theorem ocvfval 20582

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 3416	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6695	. . . . . . 7 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		ocvfval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2789	. . . . . 6 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4518	. . . . 5 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6695	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		ocvfval.i	. . . . . . . . . 10 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2789	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7208	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6695	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		ocvfval.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2789	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6699	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
15		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
16	14, 15	eqtr4di 2789	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
17	10, 16	eqeq12d 2752	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ (𝑥 , 𝑦) = 0 ))
18	17	ralbidv 3108	. . . . . 6 ⊢ (ℎ = 𝑊 → (∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ))
19	5, 18	rabeqbidv 3386	. . . . 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 5125	. . . 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 20579	. . . 4 ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
22		eqid 2736	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
23	4	fvexi 6709	. . . . . . . 8 ⊢ 𝑉 ∈ V
24		ssrab2 3979	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
25	23, 24	elpwi2 5224	. . . . . . 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 6907	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
28	23	pwex 5258	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
29		fex2 7689	. . . . 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 6796	. . 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	syl5eq 2783	1 ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  ∀wral 3051  {crab 3055  Vcvv 3398  𝒫 cpw 4499   ↦ cmpt 5120  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  Scalarcsca 16752  ·_𝑖cip 16754  0_gc0g 16898  ocvcocv 20576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-ocv 20579

This theorem is referenced by:  ocvval  20583  elocv  20584