issh3 - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > issh3		Structured version Visualization version GIF version

Theorem issh3 31023

Description: Subspace 𝐻 of a Hilbert space. (Contributed by NM, 16-Aug-1999.) (New usage is discouraged.)

**Assertion**
Ref	Expression
issh3	⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ S_ℋ ↔ (0_ℎ ∈ 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 issh3**
Step	Hyp	Ref	Expression
1		issh2 31013	. 2 ⊢ (𝐻 ∈ S_ℋ ↔ ((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)))
2		anass 468	. . 3 ⊢ (((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)) ↔ (𝐻 ⊆ ℋ ∧ (0_ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻))))
3	2	baib 535	. 2 ⊢ (𝐻 ⊆ ℋ → (((𝐻 ⊆ ℋ ∧ 0_ℎ ∈ 𝐻) ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻)) ↔ (0_ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻))))
4	1, 3	bitrid 283	1 ⊢ (𝐻 ⊆ ℋ → (𝐻 ∈ S_ℋ ↔ (0_ℎ ∈ 𝐻 ∧ (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +_ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·_ℎ 𝑦) ∈ 𝐻))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ 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: nlelshi 31864

W3C validator