Theorem shaddcl 29715

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 29707	. . . . 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 497	. . . 4 ⊢ (𝐻 ∈ Sℋ → (∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻 ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ 𝐻 (𝑥 ·ℎ 𝑦) ∈ 𝐻))
3	2	simpld 495	. . 3 ⊢ (𝐻 ∈ Sℋ → ∀𝑥 ∈ 𝐻 ∀𝑦 ∈ 𝐻 (𝑥 +ℎ 𝑦) ∈ 𝐻)
4		oveq1 7324	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 +ℎ 𝑦) = (𝐴 +ℎ 𝑦))
5	4	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 +ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 +ℎ 𝑦) ∈ 𝐻))
6		oveq2 7325	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +ℎ 𝑦) = (𝐴 +ℎ 𝐵))
7	6	eleq1d 2822	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴 +ℎ 𝑦) ∈ 𝐻 ↔ (𝐴 +ℎ 𝐵) ∈ 𝐻))
8	5, 7	rspc2v 3579	. . 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 1115	1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) → (𝐴 +ℎ 𝐵) ∈ 𝐻)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∀wral 3062   ⊆ wss 3897  (class class class)co 7317  ℂcc 10949   ℋchba 29417   +_ℎ cva 29418   ·_ℎ csm 29419  0_ℎc0v 29422   S_ℋ csh 29426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367  ax-hilex 29497  ax-hfvadd 29498  ax-hfvmul 29503

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-fv 6474  df-ov 7320  df-sh 29705

This theorem is referenced by:  shsubcl  29718  hhssabloilem  29759  hhssnv  29762  shscli  29815  shintcli  29827  shsleji  29868  shsidmi  29882  pjhthlem1  29889  spanuni  30042  spanunsni  30077  sumspansn  30147  pjaddii  30173  imaelshi  30556