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Theorem ocsh 28270
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 28267 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
2 ssrab2 3720 . . . 4 {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
31, 2syl6eqss 3688 . . 3 (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4 ssel 3630 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦𝐴𝑦 ∈ ℋ))
5 hi01 28081 . . . . . . 7 (𝑦 ∈ ℋ → (0 ·ih 𝑦) = 0)
64, 5syl6 35 . . . . . 6 (𝐴 ⊆ ℋ → (𝑦𝐴 → (0 ·ih 𝑦) = 0))
76ralrimiv 2994 . . . . 5 (𝐴 ⊆ ℋ → ∀𝑦𝐴 (0 ·ih 𝑦) = 0)
8 ax-hv0cl 27988 . . . . 5 0 ∈ ℋ
97, 8jctil 559 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0))
10 ocel 28268 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ (⊥‘𝐴) ↔ (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0)))
119, 10mpbird 247 . . 3 (𝐴 ⊆ ℋ → 0 ∈ (⊥‘𝐴))
123, 11jca 553 . 2 (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)))
13 ssel2 3631 . . . . . . . . . 10 ((𝐴 ⊆ ℋ ∧ 𝑧𝐴) → 𝑧 ∈ ℋ)
14 ax-his2 28068 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15143expa 1284 . . . . . . . . . . . . 13 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16 oveq12 6699 . . . . . . . . . . . . . 14 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17 00id 10249 . . . . . . . . . . . . . 14 (0 + 0) = 0
1816, 17syl6eq 2701 . . . . . . . . . . . . 13 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
1915, 18sylan9eq 2705 . . . . . . . . . . . 12 ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 + 𝑦) ·ih 𝑧) = 0)
2019ex 449 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2120ancoms 468 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2213, 21sylan 487 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2322an32s 863 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2423ralimdva 2991 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2524imdistanda 729 . . . . . 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 27997 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
2726anim1i 591 . . . . . 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 28268 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0)))
30 ocel 28268 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3129, 30anbi12d 747 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
32 an4 882 . . . . . . 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 3093 . . . . . . . 8 (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))
3433anbi2i 730 . . . . . . 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 267 . . . . . 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, 35syl6bb 276 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37 ocel 28268 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 + 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
3828, 36, 373imtr4d 283 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 + 𝑦) ∈ (⊥‘𝐴)))
3938ralrimivv 2999 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴))
40 mul01 10253 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41 oveq2 6698 . . . . . . . . . . . . . 14 ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
4241eqeq1d 2653 . . . . . . . . . . . . 13 ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
4340, 42syl5ibrcom 237 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4443ad2antrl 764 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45 ax-his3 28069 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
4645eqeq1d 2653 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47463expa 1284 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4847ancoms 468 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4944, 48sylibrd 249 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5013, 49sylan 487 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5150an32s 863 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5251ralimdva 2991 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5352imdistanda 729 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
54 hvmulcl 27998 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
5554anim1i 591 . . . . . 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 740 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
58 anass 682 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
5957, 58syl6bbr 278 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
60 ocel 28268 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 · 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
6156, 59, 603imtr4d 283 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 · 𝑦) ∈ (⊥‘𝐴)))
6261ralrimivv 2999 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))
6339, 62jca 553 . 2 (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴)))
64 issh2 28194 . 2 ((⊥‘𝐴) ∈ S ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))))
6512, 63, 64sylanbrc 699 1 (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974   + caddc 9977   · cmul 9979  chil 27904   + cva 27905   · csm 27906   ·ih csp 27907  0c0v 27909   S csh 27913  cort 27915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-hilex 27984  ax-hfvadd 27985  ax-hv0cl 27988  ax-hfvmul 27990  ax-hvmul0 27995  ax-hfi 28064  ax-his2 28068  ax-his3 28069
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-ltxr 10117  df-sh 28192  df-oc 28237
This theorem is referenced by:  shocsh  28271  ocss  28272  occl  28291  spanssoc  28336  ssjo  28434  chscllem2  28625
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