Theorem isch 31242

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 7439	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑m ℕ) = (𝐻 ↑m ℕ))
2	1	imaeq2d 6077	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑m ℕ)) = ( ⇝𝑣 “ (𝐻 ↑m ℕ)))
3		id 22	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 4016	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))
5		df-ch 31241	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3694	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950   “ cima 5687  (class class class)co 7432   ↑_m cmap 8867  ℕcn 12267   ⇝_𝑣 chli 30947   S_ℋ csh 30948   C_ℋ cch 30949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fv 6568  df-ov 7435  df-ch 31241

This theorem is referenced by:  isch2  31243  chsh  31244