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Theorem ocsh 29166
 Description: The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
ocsh (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )

Proof of Theorem ocsh
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ocval 29163 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
2 ssrab2 3985 . . . 4 {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
31, 2eqsstrdi 3947 . . 3 (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4 ssel 3886 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦𝐴𝑦 ∈ ℋ))
5 hi01 28979 . . . . . . 7 (𝑦 ∈ ℋ → (0 ·ih 𝑦) = 0)
64, 5syl6 35 . . . . . 6 (𝐴 ⊆ ℋ → (𝑦𝐴 → (0 ·ih 𝑦) = 0))
76ralrimiv 3113 . . . . 5 (𝐴 ⊆ ℋ → ∀𝑦𝐴 (0 ·ih 𝑦) = 0)
8 ax-hv0cl 28886 . . . . 5 0 ∈ ℋ
97, 8jctil 524 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0))
10 ocel 29164 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ (⊥‘𝐴) ↔ (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0)))
119, 10mpbird 260 . . 3 (𝐴 ⊆ ℋ → 0 ∈ (⊥‘𝐴))
123, 11jca 516 . 2 (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)))
13 ssel2 3888 . . . . . . . . . 10 ((𝐴 ⊆ ℋ ∧ 𝑧𝐴) → 𝑧 ∈ ℋ)
14 ax-his2 28966 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15143expa 1116 . . . . . . . . . . . . 13 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16 oveq12 7160 . . . . . . . . . . . . . 14 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17 00id 10854 . . . . . . . . . . . . . 14 (0 + 0) = 0
1816, 17eqtrdi 2810 . . . . . . . . . . . . 13 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
1915, 18sylan9eq 2814 . . . . . . . . . . . 12 ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 + 𝑦) ·ih 𝑧) = 0)
2019ex 417 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2120ancoms 463 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2213, 21sylan 584 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2322an32s 652 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2423ralimdva 3109 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2524imdistanda 576 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
26 hvaddcl 28895 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
2726anim1i 618 . . . . . 6 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2825, 27syl6 35 . . . . 5 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
29 ocel 29164 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0)))
30 ocel 29164 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3129, 30anbi12d 634 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
32 an4 656 . . . . . . 7 (((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
33 r19.26 3102 . . . . . . . 8 (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))
3433anbi2i 626 . . . . . . 7 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3532, 34bitr4i 281 . . . . . 6 (((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)))
3631, 35bitrdi 290 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37 ocel 29164 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 + 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
3828, 36, 373imtr4d 298 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 + 𝑦) ∈ (⊥‘𝐴)))
3938ralrimivv 3120 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴))
40 mul01 10858 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41 oveq2 7159 . . . . . . . . . . . . . 14 ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
4241eqeq1d 2761 . . . . . . . . . . . . 13 ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
4340, 42syl5ibrcom 250 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4443ad2antrl 728 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45 ax-his3 28967 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
4645eqeq1d 2761 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47463expa 1116 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4847ancoms 463 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4944, 48sylibrd 262 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5013, 49sylan 584 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5150an32s 652 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5251ralimdva 3109 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5352imdistanda 576 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
54 hvmulcl 28896 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
5554anim1i 618 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0) → ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5653, 55syl6 35 . . . . 5 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
5730anbi2d 632 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
58 anass 473 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
5957, 58bitr4di 293 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
60 ocel 29164 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 · 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
6156, 59, 603imtr4d 298 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 · 𝑦) ∈ (⊥‘𝐴)))
6261ralrimivv 3120 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))
6339, 62jca 516 . 2 (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴)))
64 issh2 29092 . 2 ((⊥‘𝐴) ∈ S ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))))
6512, 63, 64sylanbrc 587 1 (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075   ⊆ wss 3859  ‘cfv 6336  (class class class)co 7151  ℂcc 10574  0cc0 10576   + caddc 10579   · cmul 10581   ℋchba 28802   +ℎ cva 28803   ·ℎ csm 28804   ·ih csp 28805  0ℎc0v 28807   Sℋ csh 28811  ⊥cort 28813 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-hilex 28882  ax-hfvadd 28883  ax-hv0cl 28886  ax-hfvmul 28888  ax-hvmul0 28893  ax-hfi 28962  ax-his2 28966  ax-his3 28967 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-po 5444  df-so 5445  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-er 8300  df-en 8529  df-dom 8530  df-sdom 8531  df-pnf 10716  df-mnf 10717  df-ltxr 10719  df-sh 29090  df-oc 29135 This theorem is referenced by:  shocsh  29167  ocss  29168  occl  29187  spanssoc  29232  ssjo  29330  chscllem2  29521
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