Theorem shmulcl 29001

Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
shmulcl	⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻)

Proof of Theorem shmulcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issh2 28992	. . . . 5 ⊢ (𝐻 ∈ Sℋ ↔ ((𝐻 ⊆ ℋ ∧ 0ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)))
2	1	simprbi 500	. . . 4 ⊢ (𝐻 ∈ Sℋ → (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
3	2	simprd 499	. . 3 ⊢ (𝐻 ∈ Sℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)
4		oveq1 7142	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
5	4	eleq1d 2874	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝑦) ∈ 𝐻))
6		oveq2 7143	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
7	6	eleq1d 2874	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝐵) ∈ 𝐻))
8	5, 7	rspc2v 3581	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻 → (𝐴 ·ℎ 𝐵) ∈ 𝐻))
9	3, 8	syl5com 31	. 2 ⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻))
10	9	3impib 1113	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  (class class class)co 7135  ℂcc 10524   ℋchba 28702   +_ℎ cva 28703   ·_ℎ csm 28704  0_ℎc0v 28707   S_ℋ csh 28711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-hilex 28782  ax-hfvadd 28783  ax-hfvmul 28788

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-sh 28990

This theorem is referenced by:  shsubcl  29003  norm1exi  29033  hhssabloilem  29044  hhssnv  29047  shsel3  29098  shscli  29100  shintcli  29112  pjhthlem1  29174  h1de2bi  29337  h1de2ctlem  29338  spansni  29340  spansnmul  29347  spansnss  29354  spanunsni  29362  h1datomi  29364  pjmulii  29460  mayete3i  29511  imaelshi  29841  strlem1  30033  cdj1i  30216  cdj3lem1  30217