Theorem issh2 28194

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	⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))

Distinct variable group:   𝑥,𝑦,𝐻

Proof of Theorem issh2

Step	Hyp	Ref	Expression
1		issh 28193	. 2 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2		ax-hfvadd 27985	. . . . . . 7 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
3		ffun 6086	. . . . . . 7 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → Fun +ℎ )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun +ℎ
5		xpss12 5158	. . . . . . . 8 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
6	5	anidms 678	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	2	fdmi 6090	. . . . . . 7 ⊢ dom +ℎ = ( ℋ × ℋ)
8	6, 7	syl6sseqr 3685	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom +ℎ )
9		funimassov 6853	. . . . . 6 ⊢ ((Fun +ℎ ∧ (𝐻 × 𝐻) ⊆ dom +ℎ ) → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
10	4, 8, 9	sylancr 696	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
11		ax-hfvmul 27990	. . . . . . 7 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
12		ffun 6086	. . . . . . 7 ⊢ ( ·ℎ :(ℂ × ℋ)⟶ ℋ → Fun ·ℎ )
13	11, 12	ax-mp 5	. . . . . 6 ⊢ Fun ·ℎ
14		xpss2 5162	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
15	11	fdmi 6090	. . . . . . 7 ⊢ dom ·ℎ = (ℂ × ℋ)
16	14, 15	syl6sseqr 3685	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom ·ℎ )
17		funimassov 6853	. . . . . 6 ⊢ ((Fun ·ℎ ∧ (ℂ × 𝐻) ⊆ dom ·ℎ ) → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
18	13, 16, 17	sylancr 696	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
19	10, 18	anbi12d 747	. . . 4 ⊢ (𝐻 ⊆ ℋ → ((( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))
20	19	adantr 480	. . 3 ⊢ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) → ((( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻) ↔ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))
21	20	pm5.32i 670	. 2 ⊢ (((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)) ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))
22	1, 21	bitri 264	1 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))

Syntax hints:   ↔ wb 196   ∧ wa 383   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607   × cxp 5141  dom cdm 5143   “ cima 5146  Fun wfun 5920  ⟶wf 5922  (class class class)co 6690  ℂcc 9972   ℋchil 27904   +_ℎ cva 27905   ·_ℎ csm 27906  0_ℎc0v 27909   S_ℋ csh 27913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-hilex 27984  ax-hfvadd 27985  ax-hfvmul 27990

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-sh 28192

This theorem is referenced by:  shaddcl  28202  shmulcl  28203  issh3  28204  helch  28228  hsn0elch  28233  hhshsslem2  28253  ocsh  28270  shscli  28304  shintcli  28316  imaelshi  29045