| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > shmulcl | Structured version Visualization version GIF version | ||
| Description: Closure of vector scalar multiplication in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| shmulcl | ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | issh2 31228 | . . . . 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 7438 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ·ℎ 𝑦) = (𝐴 ·ℎ 𝑦)) | |
| 5 | 4 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝑦) ∈ 𝐻)) |
| 6 | oveq2 7439 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 ·ℎ 𝑦) = (𝐴 ·ℎ 𝐵)) | |
| 7 | 6 | eleq1d 2826 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 ·ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 ·ℎ 𝐵) ∈ 𝐻)) |
| 8 | 5, 7 | rspc2v 3633 | . . 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 1117 | 1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝐻) → (𝐴 ·ℎ 𝐵) ∈ 𝐻) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 (class class class)co 7431 ℂcc 11153 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 0ℎc0v 30943 Sℋ csh 30947 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 ax-hfvadd 31019 ax-hfvmul 31024 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-sh 31226 |
| This theorem is referenced by: shsubcl 31239 norm1exi 31269 hhssabloilem 31280 hhssnv 31283 shsel3 31334 shscli 31336 shintcli 31348 pjhthlem1 31410 h1de2bi 31573 h1de2ctlem 31574 spansni 31576 spansnmul 31583 spansnss 31590 spanunsni 31598 h1datomi 31600 pjmulii 31696 mayete3i 31747 imaelshi 32077 strlem1 32269 cdj1i 32452 cdj3lem1 32453 |
| Copyright terms: Public domain | W3C validator |