Theorem shaddcl 29480

Description: Closure of vector addition in a subspace of a Hilbert space. (Contributed by NM, 13-Sep-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
shaddcl	⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻)

Proof of Theorem shaddcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issh2 29472	. . . . 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	simpld 494	. . 3 ⊢ (𝐻 ∈ Sℋ → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻)
4		oveq1 7262	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ 𝑦) = (𝐴 +ℎ 𝑦))
5	4	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 +ℎ 𝑦) ∈ 𝐻))
6		oveq2 7263	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ 𝑦) = (𝐴 +ℎ 𝐵))
7	6	eleq1d 2823	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 +ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 +ℎ 𝐵) ∈ 𝐻))
8	5, 7	rspc2v 3562	. . 3 ⊢ ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 → (𝐴 +ℎ 𝐵) ∈ 𝐻))
9	3, 8	syl5com 31	. 2 ⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻))
10	9	3impib 1114	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  (class class class)co 7255  ℂcc 10800   ℋchba 29182   +_ℎ cva 29183   ·_ℎ csm 29184  0_ℎc0v 29187   S_ℋ csh 29191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-hilex 29262  ax-hfvadd 29263  ax-hfvmul 29268

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-ov 7258  df-sh 29470

This theorem is referenced by:  shsubcl  29483  hhssabloilem  29524  hhssnv  29527  shscli  29580  shintcli  29592  shsleji  29633  shsidmi  29647  pjhthlem1  29654  spanuni  29807  spanunsni  29842  sumspansn  29912  pjaddii  29938  imaelshi  30321