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Theorem issh3 31163

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 31153	. 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 206 ∧ wa 395 ∈ wcel 2109 ∀wral 3044 ⊆ wss 3903 (class class class)co 7349 ℂcc 11007 ℋchba 30863 +_ℎ cva 30864 ·_ℎ csm 30865 0_ℎc0v 30868 S_ℋ csh 30872

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pr 5371 ax-hilex 30943 ax-hfvadd 30944 ax-hfvmul 30949

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-sh 31151

This theorem is referenced by: nlelshi 32004

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