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Theorem hhsssh 31247

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 31244	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝑊 ∈ (SubSp‘𝑈))
4		shss 31188	. . 3 ⊢ (𝐻 ∈ S_ℋ → 𝐻 ⊆ ℋ)
5	3, 4	jca 511	. 2 ⊢ (𝐻 ∈ S_ℋ → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ))
6		eleq1 2819	. . 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 2731	. . . 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 5630	. . . . . . . . . . . . 13 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (𝐻 × 𝐻) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × 𝐻))
9		xpeq2 5637	. . . . . . . . . . . . 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 2766	. . . . . . . . . . . 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 5928	. . . . . . . . . . 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 5637	. . . . . . . . . . . 12 ⊢ (𝐻 = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × 𝐻) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
13	12	reseq2d 5928	. . . . . . . . . . 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 4833	. . . . . . . . . 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 5923	. . . . . . . . . 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 4833	. . . . . . . . 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 2778	. . . . . . . 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 2816	. . . . . . 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 3960	. . . . . . 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 5630	. . . . . . . . . . . 12 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → ( ℋ × ℋ) = (if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) × ℋ))
22		xpeq2 5637	. . . . . . . . . . . 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 2766	. . . . . . . . . . 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 5928	. . . . . . . . . 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 5637	. . . . . . . . . . 11 ⊢ ( ℋ = if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) → (ℂ × ℋ) = (ℂ × if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ)))
26	25	reseq2d 5928	. . . . . . . . . 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 4833	. . . . . . . . 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 5923	. . . . . . . . 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 4833	. . . . . . . 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 2816	. . . . . . 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 3960	. . . . . . 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 30978	. . . . . . . . . . . 12 ⊢ +_ℎ :( ℋ × ℋ)⟶ ℋ
34		ffn 6651	. . . . . . . . . . . 12 ⊢ ( +_ℎ :( ℋ × ℋ)⟶ ℋ → +_ℎ Fn ( ℋ × ℋ))
35		fnresdm 6600	. . . . . . . . . . . 12 ⊢ ( +_ℎ Fn ( ℋ × ℋ) → ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ )
36	33, 34, 35	mp2b 10	. . . . . . . . . . 11 ⊢ ( +_ℎ ↾ ( ℋ × ℋ)) = +_ℎ
37		ax-hfvmul 30983	. . . . . . . . . . . 12 ⊢ ·_ℎ :(ℂ × ℋ)⟶ ℋ
38		ffn 6651	. . . . . . . . . . . 12 ⊢ ( ·_ℎ :(ℂ × ℋ)⟶ ℋ → ·_ℎ Fn (ℂ × ℋ))
39		fnresdm 6600	. . . . . . . . . . . 12 ⊢ ( ·_ℎ Fn (ℂ × ℋ) → ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ )
40	37, 38, 39	mp2b 10	. . . . . . . . . . 11 ⊢ ( ·_ℎ ↾ (ℂ × ℋ)) = ·_ℎ
41	36, 40	opeq12i 4830	. . . . . . . . . 10 ⊢ ⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩ = ⟨ +_ℎ , ·_ℎ ⟩
42		normf 31101	. . . . . . . . . . 11 ⊢ norm_ℎ: ℋ⟶ℝ
43		ffn 6651	. . . . . . . . . . 11 ⊢ (norm_ℎ: ℋ⟶ℝ → norm_ℎ Fn ℋ)
44		fnresdm 6600	. . . . . . . . . . 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 4830	. . . . . . . . 9 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = ⟨⟨ +_ℎ , ·_ℎ ⟩, norm_ℎ⟩
47	46, 1	eqtr4i 2757	. . . . . . . 8 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ = 𝑈
48	1	hhnv 31143	. . . . . . . . 9 ⊢ 𝑈 ∈ NrmCVec
49		eqid 2731	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
50	49	sspid 30703	. . . . . . . . 9 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ (SubSp‘𝑈))
51	48, 50	ax-mp 5	. . . . . . . 8 ⊢ 𝑈 ∈ (SubSp‘𝑈)
52	47, 51	eqeltri 2827	. . . . . . 7 ⊢ ⟨⟨( +_ℎ ↾ ( ℋ × ℋ)), ( ·_ℎ ↾ (ℂ × ℋ))⟩, (norm_ℎ ↾ ℋ)⟩ ∈ (SubSp‘𝑈)
53		ssid 3957	. . . . . . 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 4541	. . . . 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 31246	. . 3 ⊢ if((𝑊 ∈ (SubSp‘𝑈) ∧ 𝐻 ⊆ ℋ), 𝐻, ℋ) ∈ S_ℋ
59	6, 58	dedth 4534	. 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 1541 ∈ wcel 2111 ⊆ wss 3902 ifcif 4475 ⟨cop 4582 × cxp 5614 ↾ cres 5618 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 ℂcc 11004 ℝcr 11005 NrmCVeccnv 30562 SubSpcss 30699 ℋchba 30897 +_ℎ cva 30898 ·_ℎ csm 30899 norm_ℎcno 30901 S_ℋ csh 30906

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086 ax-hilex 30977 ax-hfvadd 30978 ax-hvcom 30979 ax-hvass 30980 ax-hv0cl 30981 ax-hvaddid 30982 ax-hfvmul 30983 ax-hvmulid 30984 ax-hvmulass 30985 ax-hvdistr1 30986 ax-hvdistr2 30987 ax-hvmul0 30988 ax-hfi 31057 ax-his1 31060 ax-his2 31061 ax-his3 31062 ax-his4 31063

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-map 8752 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-icc 13252 df-seq 13909 df-exp 13969 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-topgen 17347 df-psmet 21284 df-xmet 21285 df-met 21286 df-bl 21287 df-mopn 21288 df-top 22810 df-topon 22827 df-bases 22862 df-lm 23145 df-haus 23231 df-grpo 30471 df-gid 30472 df-ginv 30473 df-gdiv 30474 df-ablo 30523 df-vc 30537 df-nv 30570 df-va 30573 df-ba 30574 df-sm 30575 df-0v 30576 df-vs 30577 df-nmcv 30578 df-ims 30579 df-ssp 30700 df-hnorm 30946 df-hba 30947 df-hvsub 30949 df-hlim 30950 df-sh 31185 df-ch 31199 df-ch0 31231

This theorem is referenced by: hhsssh2 31248

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