Theorem shmulcl 31022

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 31013	. . . . 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 496	. . . 4 ⊢ (𝐻 ∈ Sℋ → (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
3	2	simprd 495	. . 3 ⊢ (𝐻 ∈ Sℋ → ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻)
4		oveq1 7422	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦))
5	4	eleq1d 2814	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝑦) ∈ 𝐻))
6		oveq2 7423	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵))
7	6	eleq1d 2814	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝐵) ∈ 𝐻))
8	5, 7	rspc2v 3619	. . 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 1114	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057   ⊆ wss 3945  (class class class)co 7415  ℂcc 11131   ℋchba 30723   +_ℎ cva 30724   ·_ℎ csm 30725  0_ℎc0v 30728   S_ℋ csh 30732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-hilex 30803  ax-hfvadd 30804  ax-hfvmul 30809

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7418  df-sh 31011

This theorem is referenced by:  shsubcl  31024  norm1exi  31054  hhssabloilem  31065  hhssnv  31068  shsel3  31119  shscli  31121  shintcli  31133  pjhthlem1  31195  h1de2bi  31358  h1de2ctlem  31359  spansni  31361  spansnmul  31368  spansnss  31375  spanunsni  31383  h1datomi  31385  pjmulii  31481  mayete3i  31532  imaelshi  31862  strlem1  32054  cdj1i  32237  cdj3lem1  32238