Theorem ocvfval 22005

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 3475	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6911	. . . . . . 7 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		ocvfval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2816	. . . . . 6 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4577	. . . . 5 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6911	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		ocvfval.i	. . . . . . . . . 10 ⊢ , = (·𝑖‘𝑊)
9	7, 8	eqtr4di 2816	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7463	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6911	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		ocvfval.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	eqtr4di 2816	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6915	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
15		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
16	14, 15	eqtr4di 2816	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
17	10, 16	eqeq12d 2779	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ (𝑥 , 𝑦) = 0 ))
18	17	ralbidv 3189	. . . . . 6 ⊢ (ℎ = 𝑊 → (∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ))
19	5, 18	rabeqbidv 3433	. . . . 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 5196	. . . 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 22002	. . . 4 ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
22		eqid 2763	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
23	4	fvexi 6925	. . . . . . . 8 ⊢ 𝑉 ∈ V
24		ssrab2 4008	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
25	23, 24	elpwi2 5302	. . . . . . 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 7139	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
28	23	pwex 5347	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
29		fex2 7975	. . . . 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 1491	. . . 4 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) ∈ V
31	20, 21, 30	fvmpt 7019	. . 3 ⊢ (𝑊 ∈ V → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
32	2, 31	syl 18	. 2 ⊢ (𝑊 ∈ 𝑋 → (ocv‘𝑊) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))
33	1, 32	eqtrid 2810	1 ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416  Vcvv 3454  𝒫 cpw 4560   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Scalarcsca 17463  ·_𝑖cip 17465  0_gc0g 17642  ocvcocv 21999

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-ocv 22002

This theorem is used by:  ocvval  22006  elocv  22007