issh3 - Hilbert Space Explorer

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Theorem issh3 31684

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 31674	. 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 474	. . 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 545	. 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 286	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∈ wcel 2145 ∀wral 3078 ⊆ wss 3902 (class class class)co 7416 ℂcc 11123 ℋchba 31384 +_ℎ cva 31385 ·_ℎ csm 31386 0_ℎc0v 31389 S_ℋ csh 31393

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-hilex 31464 ax-hfvadd 31465 ax-hfvmul 31470

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7419 df-sh 31672

This theorem is used by: nlelshi 32525

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