Theorem ocvfval 21603

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 3490	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6900	. . . . . . 7 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		ocvfval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2785	. . . . . 6 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4621	. . . . 5 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6900	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		ocvfval.i	. . . . . . . . . 10 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2785	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7441	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6900	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		ocvfval.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2785	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6904	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
15		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
16	14, 15	eqtr4di 2785	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
17	10, 16	eqeq12d 2743	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ (𝑥 , 𝑦) = 0 ))
18	17	ralbidv 3173	. . . . . 6 ⊢ (ℎ = 𝑊 → (∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ))
19	5, 18	rabeqbidv 3446	. . . . 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 5241	. . . 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 21600	. . . 4 ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
22		eqid 2727	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
23	4	fvexi 6914	. . . . . . . 8 ⊢ 𝑉 ∈ V
24		ssrab2 4075	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
25	23, 24	elpwi2 5350	. . . . . . 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 7125	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
28	23	pwex 5382	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
29		fex2 7945	. . . . 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 1457	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
31	20, 21, 30	fvmpt 7008	. . 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 2779	1 ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ∀wral 3057  {crab 3428  Vcvv 3471  𝒫 cpw 4604   ↦ cmpt 5233  ⟶wf 6547  ‘cfv 6551  (class class class)co 7424  Basecbs 17185  Scalarcsca 17241  ·_𝑖cip 17243  0_gc0g 17426  ocvcocv 21597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-ocv 21600

This theorem is referenced by:  ocvval  21604  elocv  21605