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Theorem ocsh 27327
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 27324 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
2 ssrab2 3644 . . . 4 {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
31, 2syl6eqss 3612 . . 3 (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4 ssel 3556 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦𝐴𝑦 ∈ ℋ))
5 hi01 27138 . . . . . . 7 (𝑦 ∈ ℋ → (0 ·ih 𝑦) = 0)
64, 5syl6 34 . . . . . 6 (𝐴 ⊆ ℋ → (𝑦𝐴 → (0 ·ih 𝑦) = 0))
76ralrimiv 2942 . . . . 5 (𝐴 ⊆ ℋ → ∀𝑦𝐴 (0 ·ih 𝑦) = 0)
8 ax-hv0cl 27045 . . . . 5 0 ∈ ℋ
97, 8jctil 557 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0))
10 ocel 27325 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ (⊥‘𝐴) ↔ (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0)))
119, 10mpbird 245 . . 3 (𝐴 ⊆ ℋ → 0 ∈ (⊥‘𝐴))
123, 11jca 552 . 2 (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)))
13 ssel2 3557 . . . . . . . . . 10 ((𝐴 ⊆ ℋ ∧ 𝑧𝐴) → 𝑧 ∈ ℋ)
14 ax-his2 27125 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15143expa 1256 . . . . . . . . . . . . 13 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16 oveq12 6531 . . . . . . . . . . . . . 14 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17 00id 10057 . . . . . . . . . . . . . 14 (0 + 0) = 0
1816, 17syl6eq 2654 . . . . . . . . . . . . 13 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
1915, 18sylan9eq 2658 . . . . . . . . . . . 12 ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 + 𝑦) ·ih 𝑧) = 0)
2019ex 448 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2120ancoms 467 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2213, 21sylan 486 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2322an32s 841 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2423ralimdva 2939 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2524imdistanda 724 . . . . . 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 27054 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
2726anim1i 589 . . . . . 6 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2825, 27syl6 34 . . . . 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 27325 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0)))
30 ocel 27325 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3129, 30anbi12d 742 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
32 an4 860 . . . . . . 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 3040 . . . . . . . 8 (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))
3433anbi2i 725 . . . . . . 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 265 . . . . . 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 274 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37 ocel 27325 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 + 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
3828, 36, 373imtr4d 281 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 + 𝑦) ∈ (⊥‘𝐴)))
3938ralrimivv 2947 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴))
40 mul01 10061 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41 oveq2 6530 . . . . . . . . . . . . . 14 ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
4241eqeq1d 2606 . . . . . . . . . . . . 13 ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
4340, 42syl5ibrcom 235 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4443ad2antrl 759 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45 ax-his3 27126 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
4645eqeq1d 2606 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47463expa 1256 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4847ancoms 467 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4944, 48sylibrd 247 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5013, 49sylan 486 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5150an32s 841 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5251ralimdva 2939 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5352imdistanda 724 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
54 hvmulcl 27055 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
5554anim1i 589 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0) → ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5653, 55syl6 34 . . . . 5 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
5730anbi2d 735 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
58 anass 678 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
5957, 58syl6bbr 276 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
60 ocel 27325 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 · 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
6156, 59, 603imtr4d 281 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 · 𝑦) ∈ (⊥‘𝐴)))
6261ralrimivv 2947 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))
6339, 62jca 552 . 2 (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴)))
64 issh2 27251 . 2 ((⊥‘𝐴) ∈ S ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))))
6512, 63, 64sylanbrc 694 1 (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2890  {crab 2894  wss 3534  cfv 5785  (class class class)co 6522  cc 9785  0cc0 9787   + caddc 9790   · cmul 9792  chil 26961   + cva 26962   · csm 26963   ·ih csp 26964  0c0v 26966   S csh 26970  cort 26972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819  ax-resscn 9844  ax-1cn 9845  ax-icn 9846  ax-addcl 9847  ax-addrcl 9848  ax-mulcl 9849  ax-mulrcl 9850  ax-mulcom 9851  ax-addass 9852  ax-mulass 9853  ax-distr 9854  ax-i2m1 9855  ax-1ne0 9856  ax-1rid 9857  ax-rnegex 9858  ax-rrecex 9859  ax-cnre 9860  ax-pre-lttri 9861  ax-pre-lttrn 9862  ax-pre-ltadd 9863  ax-hilex 27041  ax-hfvadd 27042  ax-hv0cl 27045  ax-hfvmul 27047  ax-hvmul0 27052  ax-hfi 27121  ax-his2 27125  ax-his3 27126
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-nel 2777  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-po 4944  df-so 4945  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-ov 6525  df-er 7601  df-en 7814  df-dom 7815  df-sdom 7816  df-pnf 9927  df-mnf 9928  df-ltxr 9930  df-sh 27249  df-oc 27294
This theorem is referenced by:  shocsh  27328  ocss  27329  occl  27348  spanssoc  27393  ssjo  27491  chscllem2  27682
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