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Theorem issh 28912
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 28703 . . . 4 ℋ ∈ V
21elpw2 5239 . . 3 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 3anass 1087 . . 3 ((0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
42, 3anbi12i 626 . 2 ((𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
5 eleq2 2898 . . . 4 ( = 𝐻 → (0 ↔ 0𝐻))
6 id 22 . . . . . . 7 ( = 𝐻 = 𝐻)
76sqxpeqd 5580 . . . . . 6 ( = 𝐻 → ( × ) = (𝐻 × 𝐻))
87imaeq2d 5922 . . . . 5 ( = 𝐻 → ( + “ ( × )) = ( + “ (𝐻 × 𝐻)))
98, 6sseq12d 3997 . . . 4 ( = 𝐻 → (( + “ ( × )) ⊆ ↔ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻))
10 xpeq2 5569 . . . . . 6 ( = 𝐻 → (ℂ × ) = (ℂ × 𝐻))
1110imaeq2d 5922 . . . . 5 ( = 𝐻 → ( · “ (ℂ × )) = ( · “ (ℂ × 𝐻)))
1211, 6sseq12d 3997 . . . 4 ( = 𝐻 → (( · “ (ℂ × )) ⊆ ↔ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))
135, 9, 123anbi123d 1427 . . 3 ( = 𝐻 → ((0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ ) ↔ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
14 df-sh 28911 . . 3 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
1513, 14elrab2 3680 . 2 (𝐻S ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
16 anass 469 . 2 (((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
174, 15, 163bitr4i 304 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wss 3933  𝒫 cpw 4535   × cxp 5546  cima 5551  cc 10523  chba 28623   + cva 28624   · csm 28625  0c0v 28628   S csh 28632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-hilex 28703
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sh 28911
This theorem is referenced by:  issh2  28913  shss  28914  sh0  28920
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