Theorem isch 27463

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 6556	. . . 4 ⊢ (ℎ = 𝐻 → (ℎ ↑𝑚 ℕ) = (𝐻 ↑𝑚 ℕ))
2	1	imaeq2d 5385	. . 3 ⊢ (ℎ = 𝐻 → ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) = ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)))
3		id 22	. . 3 ⊢ (ℎ = 𝐻 → ℎ = 𝐻)
4	2, 3	sseq12d 3597	. 2 ⊢ (ℎ = 𝐻 → (( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ ↔ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))
5		df-ch 27462	. 2 ⊢ Cℋ = {ℎ ∈ Sℋ ∣ ( ⇝𝑣 “ (ℎ ↑𝑚 ℕ)) ⊆ ℎ}
6	4, 5	elrab2 3333	1 ⊢ (𝐻 ∈ Cℋ ↔ (𝐻 ∈ Sℋ ∧ ( ⇝𝑣 “ (𝐻 ↑𝑚 ℕ)) ⊆ 𝐻))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   “ cima 5041  (class class class)co 6549   ↑_𝑚 cmap 7744  ℕcn 10897   ⇝_𝑣 chli 27168   S_ℋ csh 27169   C_ℋ cch 27170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-ch 27462

This theorem is referenced by:  isch2  27464  chsh  27465