Theorem issh2 31542

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 31541	. 2 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (( +ℎ “ (𝐻 × 𝐻)) ⊆ 𝐻 ∧ ( ·ℎ “ (ℂ × 𝐻)) ⊆ 𝐻)))
2		ax-hfvadd 31333	. . . . . . 7 ⊢ +ℎ :( ℋ × ℋ)⟶ ℋ
3		ffun 6710	. . . . . . 7 ⊢ ( +ℎ :( ℋ × ℋ)⟶ ℋ → Fun +ℎ )
4	2, 3	ax-mp 5	. . . . . 6 ⊢ Fun +ℎ
5		xpss12 5678	. . . . . . . 8 ⊢ ((𝐻 ⊆ ℋ ∧ 𝐻 ⊆ ℋ) → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
6	5	anidms 576	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ ( ℋ × ℋ))
7	2	fdmi 6719	. . . . . . 7 ⊢ dom +ℎ = ( ℋ × ℋ)
8	6, 7	sseqtrrdi 3979	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (𝐻 × 𝐻) ⊆ dom +ℎ )
9		funimassov 7589	. . . . . 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 31338	. . . . . . 7 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ
12		ffun 6710	. . . . . . 7 ⊢ ( ·ℎ :(ℂ × ℋ)⟶ ℋ → Fun ·ℎ )
13	11, 12	ax-mp 5	. . . . . 6 ⊢ Fun ·ℎ
14		xpss2 5683	. . . . . . 7 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ (ℂ × ℋ))
15	11	fdmi 6719	. . . . . . 7 ⊢ dom ·ℎ = (ℂ × ℋ)
16	14, 15	sseqtrrdi 3979	. . . . . 6 ⊢ (𝐻 ⊆ ℋ → (ℂ × 𝐻) ⊆ dom ·ℎ )
17		funimassov 7589	. . . . . 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 2143  ∀wral 3079   ⊆ wss 3906   × cxp 5661  dom cdm 5663   “ cima 5666  Fun wfun 6532  ⟶wf 6534  (class class class)co 7412  ℂcc 11099   ℋchba 31252   +_ℎ cva 31253   ·_ℎ csm 31254  0_ℎc0v 31257   S_ℋ csh 31261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-hilex 31332  ax-hfvadd 31333  ax-hfvmul 31338

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-sh 31540

This theorem is referenced by:  shaddcl  31550  shmulcl  31551  issh3  31552  helch  31576  hsn0elch  31581  hhshsslem2  31601  ocsh  31616  shscli  31650  shintcli  31662  imaelshi  32391