HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hhsssh Structured version   Visualization version   GIF version

Theorem hhsssh 27975
Description: The predicate "𝐻 is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
hhsst.1 𝑈 = ⟨⟨ + , · ⟩, norm
hhsst.2 𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩
Assertion
Ref Expression
hhsssh (𝐻S ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

Proof of Theorem hhsssh
StepHypRef Expression
1 hhsst.1 . . . 4 𝑈 = ⟨⟨ + , · ⟩, norm
2 hhsst.2 . . . 4 𝑊 = ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩
31, 2hhsst 27972 . . 3 (𝐻S𝑊 ∈ (SubSp‘𝑈))
4 shss 27916 . . 3 (𝐻S𝐻 ⊆ ℋ)
53, 4jca 554 . 2 (𝐻S → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
6 eleq1 2686 . . 3 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻S ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S ))
7 eqid 2621 . . . 4 ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ = ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩
8 xpeq1 5088 . . . . . . . . . . . . 13 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻))
9 xpeq2 5089 . . . . . . . . . . . . 13 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
108, 9eqtrd 2655 . . . . . . . . . . . 12 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
1110reseq2d 5356 . . . . . . . . . . 11 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( + ↾ (𝐻 × 𝐻)) = ( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
12 xpeq2 5089 . . . . . . . . . . . 12 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × 𝐻) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
1312reseq2d 5356 . . . . . . . . . . 11 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( · ↾ (ℂ × 𝐻)) = ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
1411, 13opeq12d 4378 . . . . . . . . . 10 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩ = ⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩)
15 reseq2 5351 . . . . . . . . . 10 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (norm𝐻) = (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
1614, 15opeq12d 4378 . . . . . . . . 9 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨⟨( + ↾ (𝐻 × 𝐻)), ( · ↾ (ℂ × 𝐻))⟩, (norm𝐻)⟩ = ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
172, 16syl5eq 2667 . . . . . . . 8 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → 𝑊 = ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
1817eleq1d 2683 . . . . . . 7 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝑊 ∈ (SubSp‘𝑈) ↔ ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)))
19 sseq1 3605 . . . . . . 7 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 ⊆ ℋ ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ))
2018, 19anbi12d 746 . . . . . 6 (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ) ↔ (⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)))
21 xpeq1 5088 . . . . . . . . . . . 12 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ))
22 xpeq2 5089 . . . . . . . . . . . 12 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
2321, 22eqtrd 2655 . . . . . . . . . . 11 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
2423reseq2d 5356 . . . . . . . . . 10 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( + ↾ ( ℋ × ℋ)) = ( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
25 xpeq2 5089 . . . . . . . . . . 11 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × ℋ) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
2625reseq2d 5356 . . . . . . . . . 10 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( · ↾ (ℂ × ℋ)) = ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
2724, 26opeq12d 4378 . . . . . . . . 9 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩ = ⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩)
28 reseq2 5351 . . . . . . . . 9 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (norm ↾ ℋ) = (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
2927, 28opeq12d 4378 . . . . . . . 8 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ = ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
3029eleq1d 2683 . . . . . . 7 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ↔ ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)))
31 sseq1 3605 . . . . . . 7 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ ⊆ ℋ ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ))
3230, 31anbi12d 746 . . . . . 6 ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ((⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ∧ ℋ ⊆ ℋ) ↔ (⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)))
33 ax-hfvadd 27706 . . . . . . . . . . . 12 + :( ℋ × ℋ)⟶ ℋ
34 ffn 6002 . . . . . . . . . . . 12 ( + :( ℋ × ℋ)⟶ ℋ → + Fn ( ℋ × ℋ))
35 fnresdm 5958 . . . . . . . . . . . 12 ( + Fn ( ℋ × ℋ) → ( + ↾ ( ℋ × ℋ)) = + )
3633, 34, 35mp2b 10 . . . . . . . . . . 11 ( + ↾ ( ℋ × ℋ)) = +
37 ax-hfvmul 27711 . . . . . . . . . . . 12 · :(ℂ × ℋ)⟶ ℋ
38 ffn 6002 . . . . . . . . . . . 12 ( · :(ℂ × ℋ)⟶ ℋ → · Fn (ℂ × ℋ))
39 fnresdm 5958 . . . . . . . . . . . 12 ( · Fn (ℂ × ℋ) → ( · ↾ (ℂ × ℋ)) = · )
4037, 38, 39mp2b 10 . . . . . . . . . . 11 ( · ↾ (ℂ × ℋ)) = ·
4136, 40opeq12i 4375 . . . . . . . . . 10 ⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩ = ⟨ + , ·
42 normf 27829 . . . . . . . . . . 11 norm: ℋ⟶ℝ
43 ffn 6002 . . . . . . . . . . 11 (norm: ℋ⟶ℝ → norm Fn ℋ)
44 fnresdm 5958 . . . . . . . . . . 11 (norm Fn ℋ → (norm ↾ ℋ) = norm)
4542, 43, 44mp2b 10 . . . . . . . . . 10 (norm ↾ ℋ) = norm
4641, 45opeq12i 4375 . . . . . . . . 9 ⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ = ⟨⟨ + , · ⟩, norm
4746, 1eqtr4i 2646 . . . . . . . 8 ⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ = 𝑈
481hhnv 27871 . . . . . . . . 9 𝑈 ∈ NrmCVec
49 eqid 2621 . . . . . . . . . 10 (SubSp‘𝑈) = (SubSp‘𝑈)
5049sspid 27429 . . . . . . . . 9 (𝑈 ∈ NrmCVec → 𝑈 ∈ (SubSp‘𝑈))
5148, 50ax-mp 5 . . . . . . . 8 𝑈 ∈ (SubSp‘𝑈)
5247, 51eqeltri 2694 . . . . . . 7 ⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ ∈ (SubSp‘𝑈)
53 ssid 3603 . . . . . . 7 ℋ ⊆ ℋ
5452, 53pm3.2i 471 . . . . . 6 (⟨⟨( + ↾ ( ℋ × ℋ)), ( · ↾ (ℂ × ℋ))⟩, (norm ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ∧ ℋ ⊆ ℋ)
5520, 32, 54elimhyp 4118 . . . . 5 (⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)
5655simpli 474 . . . 4 ⟨⟨( + ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( · ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)
5755simpri 478 . . . 4 if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ
581, 7, 56, 57hhshsslem2 27974 . . 3 if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S
596, 58dedth 4111 . 2 ((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ) → 𝐻S )
605, 59impbii 199 1 (𝐻S ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555  ifcif 4058  cop 4154   × cxp 5072  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  cc 9878  cr 9879  NrmCVeccnv 27288  SubSpcss 27425  chil 27625   + cva 27626   · csm 27627  normcno 27629   S csh 27634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960  ax-hilex 27705  ax-hfvadd 27706  ax-hvcom 27707  ax-hvass 27708  ax-hv0cl 27709  ax-hvaddid 27710  ax-hfvmul 27711  ax-hvmulid 27712  ax-hvmulass 27713  ax-hvdistr1 27714  ax-hvdistr2 27715  ax-hvmul0 27716  ax-hfi 27785  ax-his1 27788  ax-his2 27789  ax-his3 27790  ax-his4 27791
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-sup 8292  df-inf 8293  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-n0 11237  df-z 11322  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-icc 12124  df-seq 12742  df-exp 12801  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-topgen 16025  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-top 20621  df-bases 20622  df-topon 20623  df-lm 20943  df-haus 21029  df-grpo 27196  df-gid 27197  df-ginv 27198  df-gdiv 27199  df-ablo 27248  df-vc 27263  df-nv 27296  df-va 27299  df-ba 27300  df-sm 27301  df-0v 27302  df-vs 27303  df-nmcv 27304  df-ims 27305  df-ssp 27426  df-hnorm 27674  df-hba 27675  df-hvsub 27677  df-hlim 27678  df-sh 27913  df-ch 27927  df-ch0 27959
This theorem is referenced by:  hhsssh2  27976
  Copyright terms: Public domain W3C validator