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Theorem hhsssh 31288

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 31285	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝑊 ∈ (SubSp‘𝑈))
4		shss 31229	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ)
5	3, 4	jca 511	. 2 ⊢ (𝐻 ∈ S_ℋ → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
6		eleq1 2829	. . 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 5699	. . . . . . . . . . . . 13 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻))
9		xpeq2 5706	. . . . . . . . . . . . 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 2777	. . . . . . . . . . . 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 5997	. . . . . . . . . . 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 5706	. . . . . . . . . . . 12 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × 𝐻) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
13	12	reseq2d 5997	. . . . . . . . . . 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 4881	. . . . . . . . . 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 5992	. . . . . . . . . 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 4881	. . . . . . . . 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 2789	. . . . . . . 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 2826	. . . . . . 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 4009	. . . . . . 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 632	. . . . . 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 5699	. . . . . . . . . . . 12 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ))
22		xpeq2 5706	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 5997	. . . . . . . . . 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 5706	. . . . . . . . . . 11 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × ℋ) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
26	25	reseq2d 5997	. . . . . . . . . 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 4881	. . . . . . . . 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 5992	. . . . . . . . 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 4881	. . . . . . . 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 2826	. . . . . . 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 4009	. . . . . . 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 632	. . . . . 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 31019	. . . . . . . . . . . 12 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
34		ffn 6736	. . . . . . . . . . . 12 ⊢ ( +_ℎ :( ℋ × ℋ)⟶ ℋ → +_ℎ Fn ( ℋ × ℋ))
35		fnresdm 6687	. . . . . . . . . . . 12 ⊢ ( +_ℎ Fn ( ℋ × ℋ) → ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ )
36	33, 34, 35	mp2b 10	. . . . . . . . . . 11 ⊢ ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ
37		ax-hfvmul 31024	. . . . . . . . . . . 12 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
38		ffn 6736	. . . . . . . . . . . 12 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
39		fnresdm 6687	. . . . . . . . . . . 12 ⊢ ( ·_ℎ Fn (ℂ × ℋ) → ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ )
40	37, 38, 39	mp2b 10	. . . . . . . . . . 11 ⊢ ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ
41	36, 40	opeq12i 4878	. . . . . . . . . 10 ⊢ ⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩ = ⟨ +_ℎ , ·_ℎ ⟩
42		normf 31142	. . . . . . . . . . 11 ⊢ norm_ℎ: ℋ⟶ℝ
43		ffn 6736	. . . . . . . . . . 11 ⊢ (norm_ℎ: ℋ⟶ℝ → norm_ℎ Fn ℋ)
44		fnresdm 6687	. . . . . . . . . . 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 4878	. . . . . . . . 9 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
47	46, 1	eqtr4i 2768	. . . . . . . 8 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = 𝑈
48	1	hhnv 31184	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
49		eqid 2737	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
50	49	sspid 30744	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ (SubSp‘𝑈))
51	48, 50	ax-mp 5	. . . . . . . 8 ⊢ 𝑈 ∈ (SubSp‘𝑈)
52	47, 51	eqeltri 2837	. . . . . . 7 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈)
53		ssid 4006	. . . . . . 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 4591	. . . . 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 31287	. . 3 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S_ℋ
59	6, 58	dedth 4584	. 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 1540 ∈ wcel 2108 ⊆ wss 3951 ifcif 4525 ⟨cop 4632 × cxp 5683 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ℂcc 11153 ℝcr 11154 NrmCVeccnv 30603 SubSpcss 30740 ℋchba 30938 +_ℎ cva 30939 ·_ℎ csm 30940 norm_ℎcno 30942 S_ℋ csh 30947

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-lm 23237 df-haus 23323 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-ssp 30741 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-sh 31226 df-ch 31240 df-ch0 31272

This theorem is referenced by: hhsssh2 31289

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