Theorem issh2 30956

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 30955	. 2 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2		ax-hfvadd 30747	. . . . . . 7 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
3		ffun 6711	. . . . . . 7 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → Fun +ℎ )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun +ℎ
5		xpss12 5682	. . . . . . . 8 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
6	5	anidms 566	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	2	fdmi 6720	. . . . . . 7 ⊢ dom +ℎ = ( ℋ × ℋ)
8	6, 7	sseqtrrdi 4026	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom +ℎ )
9		funimassov 7578	. . . . . 6 ⊢ ((Fun +ℎ ∧ (𝐻 × 𝐻) ⊆ dom +ℎ ) → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
10	4, 8, 9	sylancr 586	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
11		ax-hfvmul 30752	. . . . . . 7 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
12		ffun 6711	. . . . . . 7 ⊢ ( ·ℎ :(ℂ × ℋ)⟶ ℋ → Fun ·ℎ )
13	11, 12	ax-mp 5	. . . . . 6 ⊢ Fun ·ℎ
14		xpss2 5687	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
15	11	fdmi 6720	. . . . . . 7 ⊢ dom ·ℎ = (ℂ × ℋ)
16	14, 15	sseqtrrdi 4026	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom ·ℎ )
17		funimassov 7578	. . . . . 6 ⊢ ((Fun ·ℎ ∧ (ℂ × 𝐻) ⊆ dom ·ℎ ) → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
18	13, 16, 17	sylancr 586	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
19	10, 18	anbi12d 630	. . . 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 574	. 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 275	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 205   ∧ wa 395   ∈ wcel 2098  ∀wral 3053   ⊆ wss 3941   × cxp 5665  dom cdm 5667   “ cima 5670  Fun wfun 6528  ⟶wf 6530  (class class class)co 7402  ℂcc 11105   ℋchba 30666   +_ℎ cva 30667   ·_ℎ csm 30668  0_ℎc0v 30671   S_ℋ csh 30675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-hilex 30746  ax-hfvadd 30747  ax-hfvmul 30752

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-sh 30954

This theorem is referenced by:  shaddcl  30964  shmulcl  30965  issh3  30966  helch  30990  hsn0elch  30995  hhshsslem2  31015  ocsh  31030  shscli  31064  shintcli  31076  imaelshi  31805