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Theorem issh2 31228
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. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Distinct variable group:   𝑥,𝑦,𝐻

Proof of Theorem issh2
StepHypRef Expression
1 issh 31227 . 2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
2 ax-hfvadd 31019 . . . . . . 7 + :( ℋ × ℋ)⟶ ℋ
3 ffun 6739 . . . . . . 7 ( + :( ℋ × ℋ)⟶ ℋ → Fun + )
42, 3ax-mp 5 . . . . . 6 Fun +
5 xpss12 5700 . . . . . . . 8 ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
65anidms 566 . . . . . . 7 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
72fdmi 6747 . . . . . . 7 dom + = ( ℋ × ℋ)
86, 7sseqtrrdi 4025 . . . . . 6 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom + )
9 funimassov 7610 . . . . . 6 ((Fun + ∧ (𝐻 × 𝐻) ⊆ dom + ) → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
104, 8, 9sylancr 587 . . . . 5 (𝐻 ⊆ ℋ → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
11 ax-hfvmul 31024 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
12 ffun 6739 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → Fun · )
1311, 12ax-mp 5 . . . . . 6 Fun ·
14 xpss2 5705 . . . . . . 7 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
1511fdmi 6747 . . . . . . 7 dom · = (ℂ × ℋ)
1614, 15sseqtrrdi 4025 . . . . . 6 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom · )
17 funimassov 7610 . . . . . 6 ((Fun · ∧ (ℂ × 𝐻) ⊆ dom · ) → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1813, 16, 17sylancr 587 . . . . 5 (𝐻 ⊆ ℋ → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1910, 18anbi12d 632 . . . 4 (𝐻 ⊆ ℋ → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2019adantr 480 . . 3 ((𝐻 ⊆ ℋ ∧ 0𝐻) → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2120pm5.32i 574 . 2 (((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
221, 21bitri 275 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wcel 2108  wral 3061  wss 3951   × cxp 5683  dom cdm 5685  cima 5688  Fun wfun 6555  wf 6557  (class class class)co 7431  cc 11153  chba 30938   + cva 30939   · csm 30940  0c0v 30943   S csh 30947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-hilex 31018  ax-hfvadd 31019  ax-hfvmul 31024
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-sh 31226
This theorem is referenced by:  shaddcl  31236  shmulcl  31237  issh3  31238  helch  31262  hsn0elch  31267  hhshsslem2  31287  ocsh  31302  shscli  31336  shintcli  31348  imaelshi  32077
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