Theorem isch 31194

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 7348	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑m ℕ) = (𝐻 ↑m ℕ))
2	1	imaeq2d 6004	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑m ℕ)) = ( ⇝𝑣 “ (𝐻 ↑m ℕ)))
3		id 22	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 3963	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))
5		df-ch 31193	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑m ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3645	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑m ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897   “ cima 5614  (class class class)co 7341   ↑_m cmap 8745  ℕcn 12120   ⇝_𝑣 chli 30899   S_ℋ csh 30900   C_ℋ cch 30901

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-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-xp 5617  df-cnv 5619  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fv 6484  df-ov 7344  df-ch 31193

This theorem is referenced by:  isch2  31195  chsh  31196