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Theorem isch 31251
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 7438 . . . 4 ( = 𝐻 → (m ℕ) = (𝐻m ℕ))
21imaeq2d 6080 . . 3 ( = 𝐻 → ( ⇝𝑣 “ (m ℕ)) = ( ⇝𝑣 “ (𝐻m ℕ)))
3 id 22 . . 3 ( = 𝐻 = 𝐻)
42, 3sseq12d 4029 . 2 ( = 𝐻 → (( ⇝𝑣 “ (m ℕ)) ⊆ ↔ ( ⇝𝑣 “ (𝐻m ℕ)) ⊆ 𝐻))
5 df-ch 31250 . 2 C = {S ∣ ( ⇝𝑣 “ (m ℕ)) ⊆ }
64, 5elrab2 3698 1 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻m ℕ)) ⊆ 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963  cima 5692  (class class class)co 7431  m cmap 8865  cn 12264  𝑣 chli 30956   S csh 30957   C cch 30958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-ch 31250
This theorem is referenced by:  isch2  31252  chsh  31253
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