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Theorem hhsssh 31356

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 31353	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝑊 ∈ (SubSp‘𝑈))
4		shss 31297	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ)
5	3, 4	jca 511	. 2 ⊢ (𝐻 ∈ S_ℋ → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
6		eleq1 2825	. . 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 2737	. . . 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 5646	. . . . . . . . . . . . 13 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻))
9		xpeq2 5653	. . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 5946	. . . . . . . . . . 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 5653	. . . . . . . . . . . 12 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × 𝐻) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
13	12	reseq2d 5946	. . . . . . . . . . 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 4839	. . . . . . . . . 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 5941	. . . . . . . . . 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 4839	. . . . . . . . 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 2784	. . . . . . . 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 2822	. . . . . . 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 3961	. . . . . . 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 633	. . . . . 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 5646	. . . . . . . . . . . 12 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ))
22		xpeq2 5653	. . . . . . . . . . . 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 2772	. . . . . . . . . . 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 5946	. . . . . . . . . 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 5653	. . . . . . . . . . 11 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × ℋ) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
26	25	reseq2d 5946	. . . . . . . . . 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 4839	. . . . . . . . 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 5941	. . . . . . . . 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 4839	. . . . . . . 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 2822	. . . . . . 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 3961	. . . . . . 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 633	. . . . . 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 31087	. . . . . . . . . . . 12 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
34		ffn 6670	. . . . . . . . . . . 12 ⊢ ( +_ℎ :( ℋ × ℋ)⟶ ℋ → +_ℎ Fn ( ℋ × ℋ))
35		fnresdm 6619	. . . . . . . . . . . 12 ⊢ ( +_ℎ Fn ( ℋ × ℋ) → ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ )
36	33, 34, 35	mp2b 10	. . . . . . . . . . 11 ⊢ ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ
37		ax-hfvmul 31092	. . . . . . . . . . . 12 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
38		ffn 6670	. . . . . . . . . . . 12 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
39		fnresdm 6619	. . . . . . . . . . . 12 ⊢ ( ·_ℎ Fn (ℂ × ℋ) → ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ )
40	37, 38, 39	mp2b 10	. . . . . . . . . . 11 ⊢ ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ
41	36, 40	opeq12i 4836	. . . . . . . . . 10 ⊢ ⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩ = ⟨ +_ℎ , ·_ℎ ⟩
42		normf 31210	. . . . . . . . . . 11 ⊢ norm_ℎ: ℋ⟶ℝ
43		ffn 6670	. . . . . . . . . . 11 ⊢ (norm_ℎ: ℋ⟶ℝ → norm_ℎ Fn ℋ)
44		fnresdm 6619	. . . . . . . . . . 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 4836	. . . . . . . . 9 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
47	46, 1	eqtr4i 2763	. . . . . . . 8 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = 𝑈
48	1	hhnv 31252	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
49		eqid 2737	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
50	49	sspid 30812	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ (SubSp‘𝑈))
51	48, 50	ax-mp 5	. . . . . . . 8 ⊢ 𝑈 ∈ (SubSp‘𝑈)
52	47, 51	eqeltri 2833	. . . . . . 7 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈)
53		ssid 3958	. . . . . . 7 ⊢ ℋ ⊆ ℋ
54	52, 53	pm3.2i 470	. . . . . 6 ⊢ (⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈) ∧ ℋ ⊆ ℋ)
55	20, 32, 54	elimhyp 4547	. . . . 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 483	. . . 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 485	. . . 4 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ⊆ ℋ
58	1, 7, 56, 57	hhshsslem2 31355	. . 3 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S_ℋ
59	6, 58	dedth 4540	. 2 ⊢ ((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ) → 𝐻 ∈ S_ℋ )
60	5, 59	impbii 209	1 ⊢ (𝐻 ∈ S_ℋ ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ⊆ wss 3903 ifcif 4481 ⟨cop 4588 × cxp 5630 ↾ cres 5634 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 ℂcc 11036 ℝcr 11037 NrmCVeccnv 30671 SubSpcss 30808 ℋchba 31006 +_ℎ cva 31007 ·_ℎ csm 31008 norm_ℎcno 31010 S_ℋ csh 31015

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31086 ax-hfvadd 31087 ax-hvcom 31088 ax-hvass 31089 ax-hv0cl 31090 ax-hvaddid 31091 ax-hfvmul 31092 ax-hvmulid 31093 ax-hvmulass 31094 ax-hvdistr1 31095 ax-hvdistr2 31096 ax-hvmul0 31097 ax-hfi 31166 ax-his1 31169 ax-his2 31170 ax-his3 31171 ax-his4 31172

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-n0 12414 df-z 12501 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-top 22850 df-topon 22867 df-bases 22902 df-lm 23185 df-haus 23271 df-grpo 30580 df-gid 30581 df-ginv 30582 df-gdiv 30583 df-ablo 30632 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-vs 30686 df-nmcv 30687 df-ims 30688 df-ssp 30809 df-hnorm 31055 df-hba 31056 df-hvsub 31058 df-hlim 31059 df-sh 31294 df-ch 31308 df-ch0 31340

This theorem is referenced by: hhsssh2 31357

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