Theorem isch 28079

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 6657	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑𝑚 ℕ) = (𝐻 ↑𝑚 ℕ))
2	1	imaeq2d 5466	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) = ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)))
3		id 22	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 3634	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))
5		df-ch 28078	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3366	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574   “ cima 5117  (class class class)co 6650   ↑_𝑚 cmap 7857  ℕcn 11020   ⇝_𝑣 chli 27784   S_ℋ csh 27785   C_ℋ cch 27786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653  df-ch 28078

This theorem is referenced by:  isch2  28080  chsh  28081