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Theorem issh2 28996
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 28995 . 2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
2 ax-hfvadd 28787 . . . . . . 7 + :( ℋ × ℋ)⟶ ℋ
3 ffun 6494 . . . . . . 7 ( + :( ℋ × ℋ)⟶ ℋ → Fun + )
42, 3ax-mp 5 . . . . . 6 Fun +
5 xpss12 5538 . . . . . . . 8 ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
65anidms 570 . . . . . . 7 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
72fdmi 6502 . . . . . . 7 dom + = ( ℋ × ℋ)
86, 7sseqtrrdi 3969 . . . . . 6 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom + )
9 funimassov 7309 . . . . . 6 ((Fun + ∧ (𝐻 × 𝐻) ⊆ dom + ) → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
104, 8, 9sylancr 590 . . . . 5 (𝐻 ⊆ ℋ → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
11 ax-hfvmul 28792 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
12 ffun 6494 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → Fun · )
1311, 12ax-mp 5 . . . . . 6 Fun ·
14 xpss2 5543 . . . . . . 7 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
1511fdmi 6502 . . . . . . 7 dom · = (ℂ × ℋ)
1614, 15sseqtrrdi 3969 . . . . . 6 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom · )
17 funimassov 7309 . . . . . 6 ((Fun · ∧ (ℂ × 𝐻) ⊆ dom · ) → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1813, 16, 17sylancr 590 . . . . 5 (𝐻 ⊆ ℋ → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1910, 18anbi12d 633 . . . 4 (𝐻 ⊆ ℋ → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2019adantr 484 . . 3 ((𝐻 ⊆ ℋ ∧ 0𝐻) → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2120pm5.32i 578 . 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 278 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wcel 2112  wral 3109  wss 3884   × cxp 5521  dom cdm 5523  cima 5526  Fun wfun 6322  wf 6324  (class class class)co 7139  cc 10528  chba 28706   + cva 28707   · csm 28708  0c0v 28711   S csh 28715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-hilex 28786  ax-hfvadd 28787  ax-hfvmul 28792
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-sh 28994
This theorem is referenced by:  shaddcl  29004  shmulcl  29005  issh3  29006  helch  29030  hsn0elch  29035  hhshsslem2  29055  ocsh  29070  shscli  29104  shintcli  29116  imaelshi  29845
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