Theorem ocvfval 20812

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 3514	. . 3 ⊢ (𝑊 ∈ 𝑋 → 𝑊 ∈ V)
3		fveq2 6672	. . . . . . 7 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = (Base‘𝑊))
4		ocvfval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	syl6eqr 2876	. . . . . 6 ⊢ (ℎ = 𝑊 → (Base‘ℎ) = 𝑉)
6	5	pweqd 4560	. . . . 5 ⊢ (ℎ = 𝑊 → 𝒫 (Base‘ℎ) = 𝒫 𝑉)
7		fveq2 6672	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = (·𝑖‘𝑊))
8		ocvfval.i	. . . . . . . . . 10 ⊢ , = (·𝑖‘𝑊)
9	7, 8	syl6eqr 2876	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (·𝑖‘ℎ) = , )
10	9	oveqd 7175	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (𝑥(·𝑖‘ℎ)𝑦) = (𝑥 , 𝑦))
11		fveq2 6672	. . . . . . . . . . 11 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = (Scalar‘𝑊))
12		ocvfval.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
13	11, 12	syl6eqr 2876	. . . . . . . . . 10 ⊢ (ℎ = 𝑊 → (Scalar‘ℎ) = 𝐹)
14	13	fveq2d 6676	. . . . . . . . 9 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = (0g‘𝐹))
15		ocvfval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐹)
16	14, 15	syl6eqr 2876	. . . . . . . 8 ⊢ (ℎ = 𝑊 → (0g‘(Scalar‘ℎ)) = 0 )
17	10, 16	eqeq12d 2839	. . . . . . 7 ⊢ (ℎ = 𝑊 → ((𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ (𝑥 , 𝑦) = 0 ))
18	17	ralbidv 3199	. . . . . 6 ⊢ (ℎ = 𝑊 → (∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ)) ↔ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 ))
19	5, 18	rabeqbidv 3487	. . . . 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 5153	. . . 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 20809	. . . 4 ⊢ ocv = (ℎ ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘ℎ) ↦ {𝑥 ∈ (Base‘ℎ) ∣ ∀𝑦 ∈ 𝑠 (𝑥(·𝑖‘ℎ)𝑦) = (0g‘(Scalar‘ℎ))}))
22		eqid 2823	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }) = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 })
23	4	fvexi 6686	. . . . . . . 8 ⊢ 𝑉 ∈ V
24		ssrab2 4058	. . . . . . . 8 ⊢ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 } ⊆ 𝑉
25	23, 24	elpwi2 5251	. . . . . . 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 6878	. . . . 5 ⊢ (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }):𝒫 𝑉⟶𝒫 𝑉
28	23	pwex 5283	. . . . 5 ⊢ 𝒫 𝑉 ∈ V
29		fex2 7640	. . . . 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 6770	. . 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 2870	1 ⊢ (𝑊 ∈ 𝑋 → ⊥ = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑠 (𝑥 , 𝑦) = 0 }))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∀wral 3140  {crab 3144  Vcvv 3496  𝒫 cpw 4541   ↦ cmpt 5148  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Scalarcsca 16570  ·_𝑖cip 16572  0_gc0g 16715  ocvcocv 20806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-ocv 20809

This theorem is referenced by:  ocvval  20813  elocv  20814