Theorem shaddcl 28074

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 e. SH /\ A e. H /\ B e. H ) -> ( A +h B ) e. H )

Proof of Theorem shaddcl

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

Step	Hyp	Ref	Expression
1		issh2 28066	. . . . 5 |- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( A. x e. H A. y e. H ( x +h y ) e. H /\ A. x e. CC A. y e. H ( x .h y ) e. H ) ) )
2	1	simprbi 480	. . . 4 |- ( H e. SH -> ( A. x e. H A. y e. H ( x +h y ) e. H /\ A. x e. CC A. y e. H ( x .h y ) e. H ) )
3	2	simpld 475	. . 3 |- ( H e. SH -> A. x e. H A. y e. H ( x +h y ) e. H )
4		oveq1 6657	. . . . 5 |- ( x = A -> ( x +h y ) = ( A +h y ) )
5	4	eleq1d 2686	. . . 4 |- ( x = A -> ( ( x +h y ) e. H <-> ( A +h y ) e. H ) )
6		oveq2 6658	. . . . 5 |- ( y = B -> ( A +h y ) = ( A +h B ) )
7	6	eleq1d 2686	. . . 4 |- ( y = B -> ( ( A +h y ) e. H <-> ( A +h B ) e. H ) )
8	5, 7	rspc2v 3322	. . 3 |- ( ( A e. H /\ B e. H ) -> ( A. x e. H A. y e. H ( x +h y ) e. H -> ( A +h B ) e. H ) )
9	3, 8	syl5com 31	. 2 |- ( H e. SH -> ( ( A e. H /\ B e. H ) -> ( A +h B ) e. H ) )
10	9	3impib 1262	1 |- ( ( H e. SH /\ A e. H /\ B e. H ) -> ( A +h B ) e. H )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574  (class class class)co 6650   CCcc 9934   ~Hchil 27776   +h cva 27777   .h csm 27778   0hc0v 27781   SHcsh 27785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hilex 27856  ax-hfvadd 27857  ax-hfvmul 27862

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-sh 28064

This theorem is referenced by:  shsubcl  28077  hhssabloilem  28118  hhssnv  28121  shscli  28176  shintcli  28188  shsleji  28229  shsidmi  28243  pjhthlem1  28250  spanuni  28403  spanunsni  28438  sumspansn  28508  pjaddii  28534  imaelshi  28917