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Theorem issh 27449
 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 27240 . . . 4 ℋ ∈ V
21elpw2 4755 . . 3 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 3anass 1035 . . 3 ((0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
42, 3anbi12i 729 . 2 ((𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
5 eleq2 2677 . . . 4 ( = 𝐻 → (0 ↔ 0𝐻))
6 id 22 . . . . . . 7 ( = 𝐻 = 𝐻)
76sqxpeqd 5065 . . . . . 6 ( = 𝐻 → ( × ) = (𝐻 × 𝐻))
87imaeq2d 5385 . . . . 5 ( = 𝐻 → ( + “ ( × )) = ( + “ (𝐻 × 𝐻)))
98, 6sseq12d 3597 . . . 4 ( = 𝐻 → (( + “ ( × )) ⊆ ↔ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻))
10 xpeq2 5053 . . . . . 6 ( = 𝐻 → (ℂ × ) = (ℂ × 𝐻))
1110imaeq2d 5385 . . . . 5 ( = 𝐻 → ( · “ (ℂ × )) = ( · “ (ℂ × 𝐻)))
1211, 6sseq12d 3597 . . . 4 ( = 𝐻 → (( · “ (ℂ × )) ⊆ ↔ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))
135, 9, 123anbi123d 1391 . . 3 ( = 𝐻 → ((0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ ) ↔ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
14 df-sh 27448 . . 3 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
1513, 14elrab2 3333 . 2 (𝐻S ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
16 anass 679 . 2 (((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
174, 15, 163bitr4i 291 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  𝒫 cpw 4108   × cxp 5036   “ cima 5041  ℂcc 9813   ℋchil 27160   +ℎ cva 27161   ·ℎ csm 27162  0ℎc0v 27165   Sℋ csh 27169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-hilex 27240 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-sh 27448 This theorem is referenced by:  issh2  27450  shss  27451  sh0  27457
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