Theorem isch 28388

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 6820	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑𝑚 ℕ) = (𝐻 ↑𝑚 ℕ))
2	1	imaeq2d 5624	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) = ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)))
3		id 22	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 3775	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))
5		df-ch 28387	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3507	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715   “ cima 5269  (class class class)co 6813   ↑_𝑚 cmap 8023  ℕcn 11212   ⇝_𝑣 chli 28093   S_ℋ csh 28094   C_ℋ cch 28095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fv 6057  df-ov 6816  df-ch 28387

This theorem is referenced by:  isch2  28389  chsh  28390