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Theorem issh 27935
 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 27726 . . . 4 ℋ ∈ V
21elpw2 4793 . . 3 (𝐻 ∈ 𝒫 ℋ ↔ 𝐻 ⊆ ℋ)
3 3anass 1040 . . 3 ((0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
42, 3anbi12i 732 . 2 ((𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0𝐻 ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))))
5 eleq2 2687 . . . 4 ( = 𝐻 → (0 ↔ 0𝐻))
6 id 22 . . . . . . 7 ( = 𝐻 = 𝐻)
76sqxpeqd 5106 . . . . . 6 ( = 𝐻 → ( × ) = (𝐻 × 𝐻))
87imaeq2d 5430 . . . . 5 ( = 𝐻 → ( + “ ( × )) = ( + “ (𝐻 × 𝐻)))
98, 6sseq12d 3618 . . . 4 ( = 𝐻 → (( + “ ( × )) ⊆ ↔ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻))
10 xpeq2 5094 . . . . . 6 ( = 𝐻 → (ℂ × ) = (ℂ × 𝐻))
1110imaeq2d 5430 . . . . 5 ( = 𝐻 → ( · “ (ℂ × )) = ( · “ (ℂ × 𝐻)))
1211, 6sseq12d 3618 . . . 4 ( = 𝐻 → (( · “ (ℂ × )) ⊆ ↔ ( · “ (ℂ × 𝐻)) ⊆ 𝐻))
135, 9, 123anbi123d 1396 . . 3 ( = 𝐻 → ((0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ ) ↔ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
14 df-sh 27934 . . 3 S = { ∈ 𝒫 ℋ ∣ (0 ∧ ( + “ ( × )) ⊆ ∧ ( · “ (ℂ × )) ⊆ )}
1513, 14elrab2 3352 . 2 (𝐻S ↔ (𝐻 ∈ 𝒫 ℋ ∧ (0𝐻 ∧ ( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
16 anass 680 . 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 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ⊆ wss 3559  𝒫 cpw 4135   × cxp 5077   “ cima 5082  ℂcc 9886   ℋchil 27646   +ℎ cva 27647   ·ℎ csm 27648  0ℎc0v 27651   Sℋ csh 27655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-hilex 27726 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-sh 27934 This theorem is referenced by:  issh2  27936  shss  27937  sh0  27943
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