HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  issh Structured version   Visualization version   GIF version

Theorem issh 28995
Description: Subspace 𝐻 of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))

Proof of Theorem issh
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 28786 . . . 4 ℋ ∈ V
21elpw2 5215 . . 3 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 3anass 1092 . . 3 ((0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
42, 3anbi12i 629 . 2 ((𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
5 eleq2 2881 . . . 4 ( = 𝐻 → (0 ↔ 0𝐻))
6 id 22 . . . . . . 7 ( = 𝐻 = 𝐻)
76sqxpeqd 5555 . . . . . 6 ( = 𝐻 → ( × ) = (𝐻 × 𝐻))
87imaeq2d 5900 . . . . 5 ( = 𝐻 → ( + “ ( × )) = ( + “ (𝐻 × 𝐻)))
98, 6sseq12d 3951 . . . 4 ( = 𝐻 → (( + “ ( × )) ⊆ ↔ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻))
10 xpeq2 5544 . . . . . 6 ( = 𝐻 → (ℂ × ) = (ℂ × 𝐻))
1110imaeq2d 5900 . . . . 5 ( = 𝐻 → ( · “ (ℂ × )) = ( · “ (ℂ × 𝐻)))
1211, 6sseq12d 3951 . . . 4 ( = 𝐻 → (( · “ (ℂ × )) ⊆ ↔ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))
135, 9, 123anbi123d 1433 . . 3 ( = 𝐻 → ((0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ ) ↔ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
14 df-sh 28994 . . 3 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
1513, 14elrab2 3634 . 2 (𝐻S ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
16 anass 472 . 2 (((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
174, 15, 163bitr4i 306 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wss 3884  𝒫 cpw 4500   × cxp 5521  cima 5526  cc 10528  chba 28706   + cva 28707   · csm 28708  0c0v 28711   S csh 28715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-hilex 28786
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-sh 28994
This theorem is referenced by:  issh2  28996  shss  28997  sh0  29003
  Copyright terms: Public domain W3C validator