Theorem sh2 5126

Description: Subspace H of a Hilbert space.

Assertion

Ref	Expression
sh2	⊢ (H ⊆ ℋ → (H ∈ Sℋ ↔ (0v ∈ H ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))

Distinct variable group(s):   x,y,H

Proof of Theorem sh2

Step	Hyp	Ref	Expression
1		anass 336	. . 3 ⊢ (((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)) ↔ (H ⊆ ℋ ∧ (0v ∈ H ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))
2	1	baib 506	. 2 ⊢ (H ⊆ ℋ → (((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)) ↔ (0v ∈ H ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))
3		sh 5116	. 2 ⊢ (H ∈ Sℋ ↔ ((H ⊆ ℋ ∧ 0v ∈ H) ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H)))
4	2, 3	syl5bb 410	1 ⊢ (H ⊆ ℋ → (H ∈ Sℋ ↔ (0v ∈ H ∧ (∀x ∈ H ∀y ∈ H (x +v y) ∈ H ∧ ∀x ∈ ℂ ∀y ∈ H (x ·s y) ∈ H))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∀wral 1201   ⊆ wss 1487  (class class class)co 3001  ℂcc 4026   ℋ chil 4958   +_v cva 4959   ·_s csm 4960  0_vc0v 4961   S_ℋ csh 4967

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-in 1491  df-ss 1492  df-sh 5114

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