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Theorem issh2 27284
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 27283 . 2 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻)))
2 ax-hfvadd 27075 . . . . . . 7 + :( ℋ × ℋ)⟶ ℋ
3 ffun 5947 . . . . . . 7 ( + :( ℋ × ℋ)⟶ ℋ → Fun + )
42, 3ax-mp 5 . . . . . 6 Fun +
5 xpss12 5137 . . . . . . . 8 ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
65anidms 674 . . . . . . 7 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
72fdmi 5951 . . . . . . 7 dom + = ( ℋ × ℋ)
86, 7syl6sseqr 3614 . . . . . 6 (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom + )
9 funimassov 6687 . . . . . 6 ((Fun + ∧ (𝐻 × 𝐻) ⊆ dom + ) → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
104, 8, 9sylancr 693 . . . . 5 (𝐻 ⊆ ℋ → (( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻))
11 ax-hfvmul 27080 . . . . . . 7 · :(ℂ × ℋ)⟶ ℋ
12 ffun 5947 . . . . . . 7 ( · :(ℂ × ℋ)⟶ ℋ → Fun · )
1311, 12ax-mp 5 . . . . . 6 Fun ·
14 xpss2 5141 . . . . . . 7 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
1511fdmi 5951 . . . . . . 7 dom · = (ℂ × ℋ)
1614, 15syl6sseqr 3614 . . . . . 6 (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom · )
17 funimassov 6687 . . . . . 6 ((Fun · ∧ (ℂ × 𝐻) ⊆ dom · ) → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1813, 16, 17sylancr 693 . . . . 5 (𝐻 ⊆ ℋ → (( · “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻))
1910, 18anbi12d 742 . . . 4 (𝐻 ⊆ ℋ → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2019adantr 479 . . 3 ((𝐻 ⊆ ℋ ∧ 0𝐻) → ((( + “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( · “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
2120pm5.32i 666 . 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 262 1 (𝐻S ↔ ((𝐻 ⊆ ℋ ∧ 0𝐻) ∧ (∀𝑥𝐻𝑦𝐻 (𝑥 + 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦𝐻 (𝑥 · 𝑦) ∈ 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wcel 1976  wral 2895  wss 3539   × cxp 5026  dom cdm 5028  cima 5031  Fun wfun 5784  wf 5786  (class class class)co 6527  cc 9791  chil 26994   + cva 26995   · csm 26996  0c0v 26999   S csh 27003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-hilex 27074  ax-hfvadd 27075  ax-hfvmul 27080
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-sh 27282
This theorem is referenced by:  shaddcl  27292  shmulcl  27293  issh3  27294  helch  27318  hsn0elch  27323  hhshsslem2  27343  ocsh  27360  shscli  27394  shintcli  27406  imaelshi  28135
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