Theorem isch 31555

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 7419	. . . 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 31554	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3655	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3906   “ cima 5666  (class class class)co 7412   ↑_m cmap 8825  ℕcn 12234   ⇝_𝑣 chli 31260   S_ℋ csh 31261   C_ℋ cch 31262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546  df-ov 7415  df-ch 31554

This theorem is referenced by:  isch2  31556  chsh  31557