hhsssh - Hilbert Space Explorer

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

Theorem hhsssh 31750

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 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
hhsst.2	⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩

**Assertion**
Ref	Expression
hhsssh	⊢ (𝐻 ∈ S_ℋ ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

**Proof of Theorem hhsssh**
Step	Hyp	Ref	Expression
1		hhsst.1	. . . 4 ⊢ 𝑈 = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
2		hhsst.2	. . . 4 ⊢ 𝑊 = ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩
3	1, 2	hhsst 31747	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝑊 ∈ (SubSp‘𝑈))
4		shss 31691	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ)
5	3, 4	jca 521	. 2 ⊢ (𝐻 ∈ S_ℋ → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
6		eleq1 2848	. . 3 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 ∈ S_ℋ ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S_ℋ ))
7		eqid 2760	. . . 4 ⊢ ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_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))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩
8		xpeq1 5669	. . . . . . . . . . . . 13 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻))
9		xpeq2 5676	. . . . . . . . . . . . 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)))
10	8, 9	eqtrd 2795	. . . . . . . . . . . 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)))
11	10	reseq2d 5972	. . . . . . . . . . 11 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( +_ℎ ↾ (𝐻 × 𝐻)) = ( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
12		xpeq2 5676	. . . . . . . . . . . 12 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × 𝐻) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
13	12	reseq2d 5972	. . . . . . . . . . 11 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ·_ℎ ↾ (ℂ × 𝐻)) = ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
14	11, 13	opeq12d 4841	. . . . . . . . . 10 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩ = ⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩)
15		reseq2 5967	. . . . . . . . . 10 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (norm_ℎ ↾ 𝐻) = (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
16	14, 15	opeq12d 4841	. . . . . . . . 9 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨⟨( +_ℎ ↾ (𝐻 × 𝐻)), ( ·_ℎ ↾ (ℂ × 𝐻))⟩, (norm_ℎ ↾ 𝐻)⟩ = ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
17	2, 16	eqtrid 2807	. . . . . . . 8 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → 𝑊 = ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
18	17	eleq1d 2845	. . . . . . 7 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝑊 ∈ (SubSp‘𝑈) ↔ ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)))
19		sseq1 3956	. . . . . . 7 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 ⊆ ℋ ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ))
20	18, 19	anbi12d 644	. . . . . 6 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ) ↔ (⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)))
21		xpeq1 5669	. . . . . . . . . . . 12 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ))
22		xpeq2 5676	. . . . . . . . . . . 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)))
23	21, 22	eqtrd 2795	. . . . . . . . . . 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)))
24	23	reseq2d 5972	. . . . . . . . . 10 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( +_ℎ ↾ ( ℋ × ℋ)) = ( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
25		xpeq2 5676	. . . . . . . . . . 11 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × ℋ) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
26	25	reseq2d 5972	. . . . . . . . . 10 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ·_ℎ ↾ (ℂ × ℋ)) = ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))))
27	24, 26	opeq12d 4841	. . . . . . . . 9 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩ = ⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩)
28		reseq2 5967	. . . . . . . . 9 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (norm_ℎ ↾ ℋ) = (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
29	27, 28	opeq12d 4841	. . . . . . . 8 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩)
30	29	eleq1d 2845	. . . . . . 7 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ↔ ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)))
31		sseq1 3956	. . . . . . 7 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ ⊆ ℋ ↔ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ))
32	30, 31	anbi12d 644	. . . . . 6 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ((⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ∧ ℋ ⊆ ℋ) ↔ (⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)))
33		ax-hfvadd 31481	. . . . . . . . . . . 12 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
34		ffn 6702	. . . . . . . . . . . 12 ⊢ ( +_ℎ :( ℋ × ℋ)⟶ ℋ → +_ℎ Fn ( ℋ × ℋ))
35		fnresdm 6651	. . . . . . . . . . . 12 ⊢ ( +_ℎ Fn ( ℋ × ℋ) → ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ )
36	33, 34, 35	mp2b 10	. . . . . . . . . . 11 ⊢ ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ
37		ax-hfvmul 31486	. . . . . . . . . . . 12 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
38		ffn 6702	. . . . . . . . . . . 12 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
39		fnresdm 6651	. . . . . . . . . . . 12 ⊢ ( ·_ℎ Fn (ℂ × ℋ) → ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ )
40	37, 38, 39	mp2b 10	. . . . . . . . . . 11 ⊢ ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ
41	36, 40	opeq12i 4838	. . . . . . . . . 10 ⊢ ⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩ = ⟨ +_ℎ , ·_ℎ ⟩
42		normf 31604	. . . . . . . . . . 11 ⊢ norm_ℎ: ℋ⟶ℝ
43		ffn 6702	. . . . . . . . . . 11 ⊢ (norm_ℎ: ℋ⟶ℝ → norm_ℎ Fn ℋ)
44		fnresdm 6651	. . . . . . . . . . 11 ⊢ (norm_ℎ Fn ℋ → (norm_ℎ ↾ ℋ) = norm_ℎ)
45	42, 43, 44	mp2b 10	. . . . . . . . . 10 ⊢ (norm_ℎ ↾ ℋ) = norm_ℎ
46	41, 45	opeq12i 4838	. . . . . . . . 9 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
47	46, 1	eqtr4i 2786	. . . . . . . 8 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = 𝑈
48	1	hhnv 31646	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
49		eqid 2760	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
50	49	sspid 31206	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ (SubSp‘𝑈))
51	48, 50	ax-mp 5	. . . . . . . 8 ⊢ 𝑈 ∈ (SubSp‘𝑈)
52	47, 51	eqeltri 2856	. . . . . . 7 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈)
53		ssid 3953	. . . . . . 7 ⊢ ℋ ⊆ ℋ
54	52, 53	pm3.2i 476	. . . . . 6 ⊢ (⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ∧ ℋ ⊆ ℋ)
55	20, 32, 54	elimhyp 4548	. . . . 5 ⊢ (⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈) ∧ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ)
56	55	simpli 489	. . . 4 ⊢ ⟨⟨( +_ℎ ↾ (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))), ( ·_ℎ ↾ (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))⟩, (norm_ℎ ↾ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ))⟩ ∈ (SubSp‘𝑈)
57	55	simpri 491	. . . 4 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ
58	1, 7, 56, 57	hhshsslem2 31749	. . 3 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S_ℋ
59	6, 58	dedth 4541	. 2 ⊢ ((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ) → 𝐻 ∈ S_ℋ )
60	5, 59	impbii 212	1 ⊢ (𝐻 ∈ S_ℋ ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 ifcif 4482 ⟨cop 4590 × cxp 5653 ↾ cres 5657 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 ℂcc 11122 ℝcr 11123 NrmCVeccnv 31065 SubSpcss 31202 ℋchba 31400 +_ℎ cva 31401 ·_ℎ csm 31402 norm_ℎcno 31404 S_ℋ csh 31409

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204 ax-hilex 31480 ax-hfvadd 31481 ax-hvcom 31482 ax-hvass 31483 ax-hv0cl 31484 ax-hvaddid 31485 ax-hfvmul 31486 ax-hvmulid 31487 ax-hvmulass 31488 ax-hvdistr1 31489 ax-hvdistr2 31490 ax-hvmul0 31491 ax-hfi 31560 ax-his1 31563 ax-his2 31564 ax-his3 31565 ax-his4 31566

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-n0 12529 df-z 12616 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-icc 13405 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-topgen 17528 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-top 23119 df-topon 23136 df-bases 23171 df-lm 23454 df-haus 23540 df-grpo 30974 df-gid 30975 df-ginv 30976 df-gdiv 30977 df-ablo 31026 df-vc 31040 df-nv 31073 df-va 31076 df-ba 31077 df-sm 31078 df-0v 31079 df-vs 31080 df-nmcv 31081 df-ims 31082 df-ssp 31203 df-hnorm 31449 df-hba 31450 df-hvsub 31452 df-hlim 31453 df-sh 31688 df-ch 31702 df-ch0 31734

This theorem is used by: hhsssh2 31751

W3C validator