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Theorem issh 28193
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 27984 . . . 4 ℋ ∈ V
21elpw2 4858 . . 3 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 3anass 1059 . . 3 ((0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
42, 3anbi12i 733 . 2 ((𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
5 eleq2 2719 . . . 4 ( = 𝐻 → (0 ↔ 0𝐻))
6 id 22 . . . . . . 7 ( = 𝐻 = 𝐻)
76sqxpeqd 5175 . . . . . 6 ( = 𝐻 → ( × ) = (𝐻 × 𝐻))
87imaeq2d 5501 . . . . 5 ( = 𝐻 → ( + “ ( × )) = ( + “ (𝐻 × 𝐻)))
98, 6sseq12d 3667 . . . 4 ( = 𝐻 → (( + “ ( × )) ⊆ ↔ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻))
10 xpeq2 5163 . . . . . 6 ( = 𝐻 → (ℂ × ) = (ℂ × 𝐻))
1110imaeq2d 5501 . . . . 5 ( = 𝐻 → ( · “ (ℂ × )) = ( · “ (ℂ × 𝐻)))
1211, 6sseq12d 3667 . . . 4 ( = 𝐻 → (( · “ (ℂ × )) ⊆ ↔ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))
135, 9, 123anbi123d 1439 . . 3 ( = 𝐻 → ((0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ ) ↔ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
14 df-sh 28192 . . 3 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
1513, 14elrab2 3399 . 2 (𝐻S ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
16 anass 682 . 2 (((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
174, 15, 163bitr4i 292 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  𝒫 cpw 4191   × cxp 5141  cima 5146  cc 9972  chil 27904   + cva 27905   · csm 27906  0c0v 27909   S csh 27913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-hilex 27984
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-sh 28192
This theorem is referenced by:  issh2  28194  shss  28195  sh0  28201
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