Theorem isch 31621

Description: Closed subspace 𝐻 of a Hilbert space. (Contributed by NM, 17-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
isch	⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))

Proof of Theorem isch

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7423	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑m ℕ) = (𝐻 ↑m ℕ))
2	1	imaeq2d 6064	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑m ℕ)) = ( ⇝𝑣 “ (𝐻 ↑m ℕ)))
3		id 23	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 3971	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))
5		df-ch 31620	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3656	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906   “ cima 5666  (class class class)co 7416   ↑_m cmap 8826  ℕcn 12244   ⇝_𝑣 chli 31326   S_ℋ csh 31327   C_ℋ cch 31328

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 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fv 6548  df-ov 7419  df-ch 31620

This theorem is used by:  isch2  31622  chsh  31623