Theorem issh2 31502

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 31501	. 2 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2		ax-hfvadd 31293	. . . . . . 7 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
3		ffun 6709	. . . . . . 7 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → Fun +ℎ )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun +ℎ
5		xpss12 5677	. . . . . . . 8 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
6	5	anidms 576	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	2	fdmi 6718	. . . . . . 7 ⊢ dom +ℎ = ( ℋ × ℋ)
8	6, 7	sseqtrrdi 3986	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom +ℎ )
9		funimassov 7588	. . . . . 6 ⊢ ((Fun +ℎ ∧ (𝐻 × 𝐻) ⊆ dom +ℎ ) → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
10	4, 8, 9	sylancr 598	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻))
11		ax-hfvmul 31298	. . . . . . 7 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
12		ffun 6709	. . . . . . 7 ⊢ ( ·ℎ :(ℂ × ℋ)⟶ ℋ → Fun ·ℎ )
13	11, 12	ax-mp 5	. . . . . 6 ⊢ Fun ·ℎ
14		xpss2 5682	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
15	11	fdmi 6718	. . . . . . 7 ⊢ dom ·ℎ = (ℂ × ℋ)
16	14, 15	sseqtrrdi 3986	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom ·ℎ )
17		funimassov 7588	. . . . . 6 ⊢ ((Fun ·ℎ ∧ (ℂ × 𝐻) ⊆ dom ·ℎ ) → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
18	13, 16, 17	sylancr 598	. . . . 5 ⊢ (𝐻 ⊆ ℋ → (( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻 ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
19	10, 18	anbi12d 643	. . . 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 485	. . 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 584	. 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 278	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 209   ∧ wa 400   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913   × cxp 5660  dom cdm 5662   “ cima 5665  Fun wfun 6531  ⟶wf 6533  (class class class)co 7411  ℂcc 11098   ℋchba 31212   +_ℎ cva 31213   ·_ℎ csm 31214  0_ℎc0v 31217   S_ℋ csh 31221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-hilex 31292  ax-hfvadd 31293  ax-hfvmul 31298

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-sh 31500

This theorem is referenced by:  shaddcl  31510  shmulcl  31511  issh3  31512  helch  31536  hsn0elch  31541  hhshsslem2  31561  ocsh  31576  shscli  31610  shintcli  31622  imaelshi  32351