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Theorem issh2 28669
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 28668 . 2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
2 ax-hfvadd 28460 . . . . . . 7 + :( ℋ × ℋ)⟶ ℋ
3 ffun 6388 . . . . . . 7 ( + :( ℋ × ℋ)⟶ ℋ → Fun + )
42, 3ax-mp 5 . . . . . 6 Fun +
5 xpss12 5461 . . . . . . . 8 ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
65anidms 567 . . . . . . 7 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
72fdmi 6395 . . . . . . 7 dom + = ( ℋ × ℋ)
86, 7syl6sseqr 3941 . . . . . 6 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom + )
9 funimassov 7184 . . . . . 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 28465 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
12 ffun 6388 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → Fun · )
1311, 12ax-mp 5 . . . . . 6 Fun ·
14 xpss2 5466 . . . . . . 7 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
1511fdmi 6395 . . . . . . 7 dom · = (ℂ × ℋ)
1614, 15syl6sseqr 3941 . . . . . 6 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom · )
17 funimassov 7184 . . . . . 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 630 . . . 4 (𝐻 ⊆ ℋ → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2019adantr 481 . . 3 ((𝐻 ⊆ ℋ ∧ 0𝐻) → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2120pm5.32i 575 . 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 276 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2080  wral 3104  wss 3861   × cxp 5444  dom cdm 5446  cima 5449  Fun wfun 6222  wf 6224  (class class class)co 7019  cc 10384  chba 28379   + cva 28380   · csm 28381  0c0v 28384   S csh 28388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pr 5224  ax-hilex 28459  ax-hfvadd 28460  ax-hfvmul 28465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-pw 4457  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-ov 7022  df-sh 28667
This theorem is referenced by:  shaddcl  28677  shmulcl  28678  issh3  28679  helch  28703  hsn0elch  28708  hhshsslem2  28728  ocsh  28743  shscli  28777  shintcli  28789  imaelshi  29518
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