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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 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))

Proof of Theorem isch
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oveq1 6820 . . . 4 ( = 𝐻 → (𝑚 ℕ) = (𝐻𝑚 ℕ))
21imaeq2d 5624 . . 3 ( = 𝐻 → ( ⇝𝑣 “ (𝑚 ℕ)) = ( ⇝𝑣 “ (𝐻𝑚 ℕ)))
3 id 22 . . 3 ( = 𝐻 = 𝐻)
42, 3sseq12d 3775 . 2 ( = 𝐻 → (( ⇝𝑣 “ (𝑚 ℕ)) ⊆ ↔ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
5 df-ch 28387 . 2 C = {S ∣ ( ⇝𝑣 “ (𝑚 ℕ)) ⊆ }
64, 5elrab2 3507 1 (𝐻C ↔ (𝐻S ∧ ( ⇝𝑣 “ (𝐻𝑚 ℕ)) ⊆ 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wss 3715  cima 5269  (class class class)co 6813  𝑚 cmap 8023  cn 11212  𝑣 chli 28093   S csh 28094   C 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
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