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Theorem issh2 28346
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 28345 . 2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
2 ax-hfvadd 28137 . . . . . . 7 + :( ℋ × ℋ)⟶ ℋ
3 ffun 6197 . . . . . . 7 ( + :( ℋ × ℋ)⟶ ℋ → Fun + )
42, 3ax-mp 5 . . . . . 6 Fun +
5 xpss12 5269 . . . . . . . 8 ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
65anidms 680 . . . . . . 7 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
72fdmi 6201 . . . . . . 7 dom + = ( ℋ × ℋ)
86, 7syl6sseqr 3781 . . . . . 6 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom + )
9 funimassov 6964 . . . . . 6 ((Fun + ∧ (𝐻 × 𝐻) ⊆ dom + ) → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
104, 8, 9sylancr 698 . . . . 5 (𝐻 ⊆ ℋ → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
11 ax-hfvmul 28142 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
12 ffun 6197 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → Fun · )
1311, 12ax-mp 5 . . . . . 6 Fun ·
14 xpss2 5273 . . . . . . 7 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
1511fdmi 6201 . . . . . . 7 dom · = (ℂ × ℋ)
1614, 15syl6sseqr 3781 . . . . . 6 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom · )
17 funimassov 6964 . . . . . 6 ((Fun · ∧ (ℂ × 𝐻) ⊆ dom · ) → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1813, 16, 17sylancr 698 . . . . 5 (𝐻 ⊆ ℋ → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1910, 18anbi12d 749 . . . 4 (𝐻 ⊆ ℋ → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2019adantr 472 . . 3 ((𝐻 ⊆ ℋ ∧ 0𝐻) → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2120pm5.32i 672 . 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 264 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2127  wral 3038  wss 3703   × cxp 5252  dom cdm 5254  cima 5257  Fun wfun 6031  wf 6033  (class class class)co 6801  cc 10097  chil 28056   + cva 28057   · csm 28058  0c0v 28061   S csh 28065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-hilex 28136  ax-hfvadd 28137  ax-hfvmul 28142
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-sh 28344
This theorem is referenced by:  shaddcl  28354  shmulcl  28355  issh3  28356  helch  28380  hsn0elch  28385  hhshsslem2  28405  ocsh  28422  shscli  28456  shintcli  28468  imaelshi  29197
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