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Theorem ocsh 28987
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 28984 . . . 4 (𝐴 ⊆ ℋ → (⊥‘𝐴) = {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0})
2 ssrab2 4053 . . . 4 {𝑥 ∈ ℋ ∣ ∀𝑦𝐴 (𝑥 ·ih 𝑦) = 0} ⊆ ℋ
31, 2eqsstrdi 4018 . . 3 (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
4 ssel 3958 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦𝐴𝑦 ∈ ℋ))
5 hi01 28800 . . . . . . 7 (𝑦 ∈ ℋ → (0 ·ih 𝑦) = 0)
64, 5syl6 35 . . . . . 6 (𝐴 ⊆ ℋ → (𝑦𝐴 → (0 ·ih 𝑦) = 0))
76ralrimiv 3178 . . . . 5 (𝐴 ⊆ ℋ → ∀𝑦𝐴 (0 ·ih 𝑦) = 0)
8 ax-hv0cl 28707 . . . . 5 0 ∈ ℋ
97, 8jctil 520 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0))
10 ocel 28985 . . . 4 (𝐴 ⊆ ℋ → (0 ∈ (⊥‘𝐴) ↔ (0 ∈ ℋ ∧ ∀𝑦𝐴 (0 ·ih 𝑦) = 0)))
119, 10mpbird 258 . . 3 (𝐴 ⊆ ℋ → 0 ∈ (⊥‘𝐴))
123, 11jca 512 . 2 (𝐴 ⊆ ℋ → ((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)))
13 ssel2 3959 . . . . . . . . . 10 ((𝐴 ⊆ ℋ ∧ 𝑧𝐴) → 𝑧 ∈ ℋ)
14 ax-his2 28787 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
15143expa 1110 . . . . . . . . . . . . 13 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 + 𝑦) ·ih 𝑧) = ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)))
16 oveq12 7154 . . . . . . . . . . . . . 14 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = (0 + 0))
17 00id 10803 . . . . . . . . . . . . . 14 (0 + 0) = 0
1816, 17syl6eq 2869 . . . . . . . . . . . . 13 (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 ·ih 𝑧) + (𝑦 ·ih 𝑧)) = 0)
1915, 18sylan9eq 2873 . . . . . . . . . . . 12 ((((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) ∧ ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0)) → ((𝑥 + 𝑦) ·ih 𝑧) = 0)
2019ex 413 . . . . . . . . . . 11 (((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2120ancoms 459 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2213, 21sylan 580 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2322an32s 648 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → (((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2423ralimdva 3174 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) → ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0))
2524imdistanda 572 . . . . . 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 28716 . . . . . . 7 ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) → (𝑥 + 𝑦) ∈ ℋ)
2726anim1i 614 . . . . . 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 28985 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑥 ∈ (⊥‘𝐴) ↔ (𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0)))
30 ocel 28985 . . . . . . 7 (𝐴 ⊆ ℋ → (𝑦 ∈ (⊥‘𝐴) ↔ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
3129, 30anbi12d 630 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ ∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0) ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
32 an4 652 . . . . . . 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 3167 . . . . . . . 8 (∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0) ↔ (∀𝑧𝐴 (𝑥 ·ih 𝑧) = 0 ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))
3433anbi2i 622 . . . . . . 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 279 . . . . . 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 288 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℋ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 ·ih 𝑧) = 0 ∧ (𝑦 ·ih 𝑧) = 0))))
37 ocel 28985 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 + 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 + 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 + 𝑦) ·ih 𝑧) = 0)))
3828, 36, 373imtr4d 295 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ (⊥‘𝐴) ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 + 𝑦) ∈ (⊥‘𝐴)))
3938ralrimivv 3187 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴))
40 mul01 10807 . . . . . . . . . . . . 13 (𝑥 ∈ ℂ → (𝑥 · 0) = 0)
41 oveq2 7153 . . . . . . . . . . . . . 14 ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = (𝑥 · 0))
4241eqeq1d 2820 . . . . . . . . . . . . 13 ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · (𝑦 ·ih 𝑧)) = 0 ↔ (𝑥 · 0) = 0))
4340, 42syl5ibrcom 248 . . . . . . . . . . . 12 (𝑥 ∈ ℂ → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4443ad2antrl 724 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → (𝑥 · (𝑦 ·ih 𝑧)) = 0))
45 ax-his3 28788 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 · 𝑦) ·ih 𝑧) = (𝑥 · (𝑦 ·ih 𝑧)))
4645eqeq1d 2820 . . . . . . . . . . . . 13 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
47463expa 1110 . . . . . . . . . . . 12 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4847ancoms 459 . . . . . . . . . . 11 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (((𝑥 · 𝑦) ·ih 𝑧) = 0 ↔ (𝑥 · (𝑦 ·ih 𝑧)) = 0))
4944, 48sylibrd 260 . . . . . . . . . 10 ((𝑧 ∈ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5013, 49sylan 580 . . . . . . . . 9 (((𝐴 ⊆ ℋ ∧ 𝑧𝐴) ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5150an32s 648 . . . . . . . 8 (((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) ∧ 𝑧𝐴) → ((𝑦 ·ih 𝑧) = 0 → ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5251ralimdva 3174 . . . . . . 7 ((𝐴 ⊆ ℋ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ)) → (∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0 → ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0))
5352imdistanda 572 . . . . . 6 (𝐴 ⊆ ℋ → (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
54 hvmulcl 28717 . . . . . . 7 ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · 𝑦) ∈ ℋ)
5554anim1i 614 . . . . . 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 628 . . . . . 6 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0))))
58 anass 469 . . . . . 6 (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0) ↔ (𝑥 ∈ ℂ ∧ (𝑦 ∈ ℋ ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
5957, 58syl6bbr 290 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) ↔ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ ∀𝑧𝐴 (𝑦 ·ih 𝑧) = 0)))
60 ocel 28985 . . . . 5 (𝐴 ⊆ ℋ → ((𝑥 · 𝑦) ∈ (⊥‘𝐴) ↔ ((𝑥 · 𝑦) ∈ ℋ ∧ ∀𝑧𝐴 ((𝑥 · 𝑦) ·ih 𝑧) = 0)))
6156, 59, 603imtr4d 295 . . . 4 (𝐴 ⊆ ℋ → ((𝑥 ∈ ℂ ∧ 𝑦 ∈ (⊥‘𝐴)) → (𝑥 · 𝑦) ∈ (⊥‘𝐴)))
6261ralrimivv 3187 . . 3 (𝐴 ⊆ ℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))
6339, 62jca 512 . 2 (𝐴 ⊆ ℋ → (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴)))
64 issh2 28913 . 2 ((⊥‘𝐴) ∈ S ↔ (((⊥‘𝐴) ⊆ ℋ ∧ 0 ∈ (⊥‘𝐴)) ∧ (∀𝑥 ∈ (⊥‘𝐴)∀𝑦 ∈ (⊥‘𝐴)(𝑥 + 𝑦) ∈ (⊥‘𝐴) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ (⊥‘𝐴)(𝑥 · 𝑦) ∈ (⊥‘𝐴))))
6512, 63, 64sylanbrc 583 1 (𝐴 ⊆ ℋ → (⊥‘𝐴) ∈ S )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  {crab 3139  wss 3933  cfv 6348  (class class class)co 7145  cc 10523  0cc0 10525   + caddc 10528   · cmul 10530  chba 28623   + cva 28624   · csm 28625   ·ih csp 28626  0c0v 28628   S csh 28632  cort 28634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-hilex 28703  ax-hfvadd 28704  ax-hv0cl 28707  ax-hfvmul 28709  ax-hvmul0 28714  ax-hfi 28783  ax-his2 28787  ax-his3 28788
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-ltxr 10668  df-sh 28911  df-oc 28956
This theorem is referenced by:  shocsh  28988  ocss  28989  occl  29008  spanssoc  29053  ssjo  29151  chscllem2  29342
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