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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 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))

Proof of Theorem isch
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq1 6556 . . . 4 ( = 𝐻 → (𝑚 ℕ) = (𝐻𝑚 ℕ))
21imaeq2d 5385 . . 3 ( = 𝐻 → ( ⇝𝑣 “ (𝑚 ℕ)) = ( ⇝𝑣 “ (𝐻𝑚 ℕ)))
3 id 22 . . 3 ( = 𝐻 = 𝐻)
42, 3sseq12d 3597 . 2 ( = 𝐻 → (( ⇝𝑣 “ (𝑚 ℕ)) ⊆ ↔ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
5 df-ch 27462 . 2 C = {S ∣ ( ⇝𝑣 “ (𝑚 ℕ)) ⊆ }
64, 5elrab2 3333 1 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
 Colors of variables: wff setvar class 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   Sℋ csh 27169   Cℋ 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
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