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Theorem ocsh 28451
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 28448 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
2 ssrab2 3828 . . . 4 {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
31, 2syl6eqss 3796 . . 3 (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4 ssel 3738 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦𝐴𝑦 ∈ ℋ))
5 hi01 28262 . . . . . . 7 (𝑦 ∈ ℋ → (0 ·ih 𝑦) = 0)
64, 5syl6 35 . . . . . 6 (𝐴 ⊆ ℋ → (𝑦𝐴 → (0 ·ih 𝑦) = 0))
76ralrimiv 3103 . . . . 5 (𝐴 ⊆ ℋ → ∀𝑦𝐴 (0 ·ih 𝑦) = 0)
8 ax-hv0cl 28169 . . . . 5 0 ∈ ℋ
97, 8jctil 561 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0))
10 ocel 28449 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ (⊥‘𝐴) ↔ (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0)))
119, 10mpbird 247 . . 3 (𝐴 ⊆ ℋ → 0 ∈ (⊥‘𝐴))
123, 11jca 555 . 2 (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)))
13 ssel2 3739 . . . . . . . . . 10 ((𝐴 ⊆ ℋ ∧ 𝑧𝐴) → 𝑧 ∈ ℋ)
14 ax-his2 28249 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15143expa 1112 . . . . . . . . . . . . 13 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16 oveq12 6822 . . . . . . . . . . . . . 14 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17 00id 10403 . . . . . . . . . . . . . 14 (0 + 0) = 0
1816, 17syl6eq 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 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 489 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2322an32s 881 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2423ralimdva 3100 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2524imdistanda 731 . . . . . 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 28178 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
2726anim1i 593 . . . . . 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 28449 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0)))
30 ocel 28449 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3129, 30anbi12d 749 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
32 an4 900 . . . . . . 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 3202 . . . . . . . 8 (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))
3433anbi2i 732 . . . . . . 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 28449 . . . . 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 3108 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴))
40 mul01 10407 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41 oveq2 6821 . . . . . . . . . . . . . 14 ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
4241eqeq1d 2762 . . . . . . . . . . . . 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 766 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45 ax-his3 28250 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
4645eqeq1d 2762 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47463expa 1112 . . . . . . . . . . . 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 489 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5150an32s 881 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5251ralimdva 3100 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5352imdistanda 731 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
54 hvmulcl 28179 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
5554anim1i 593 . . . . . 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 742 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
58 anass 684 . . . . . 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 28449 . . . . 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 3108 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))
6339, 62jca 555 . 2 (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴)))
64 issh2 28375 . 2 ((⊥‘𝐴) ∈ S ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))))
6512, 63, 64sylanbrc 701 1 (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  {crab 3054  wss 3715  cfv 6049  (class class class)co 6813  cc 10126  0cc0 10128   + caddc 10131   · cmul 10133  chil 28085   + cva 28086   · csm 28087   ·ih csp 28088  0c0v 28090   S csh 28094  cort 28096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-hilex 28165  ax-hfvadd 28166  ax-hv0cl 28169  ax-hfvmul 28171  ax-hvmul0 28176  ax-hfi 28245  ax-his2 28249  ax-his3 28250
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-er 7911  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-ltxr 10271  df-sh 28373  df-oc 28418
This theorem is referenced by:  shocsh  28452  ocss  28453  occl  28472  spanssoc  28517  ssjo  28615  chscllem2  28806
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