Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  issh Structured version   Visualization version   Unicode version

Theorem issh 26942
 Description: Subspace of a Hilbert space. A subspace is a subset of Hilbert space which contains the zero vector and is closed under vector addition and scalar multiplication. (Contributed by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
issh

Proof of Theorem issh
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 26733 . . . 4
21elpw2 4565 . . 3
3 3anass 1011 . . 3
42, 3anbi12i 711 . 2
5 eleq2 2538 . . . 4
6 id 22 . . . . . . 7
76sqxpeqd 4865 . . . . . 6
87imaeq2d 5174 . . . . 5
98, 6sseq12d 3447 . . . 4
10 xpeq2 4854 . . . . . 6
1110imaeq2d 5174 . . . . 5
1211, 6sseq12d 3447 . . . 4
135, 9, 123anbi123d 1365 . . 3
14 df-sh 26941 . . 3
1513, 14elrab2 3186 . 2
16 anass 661 . 2
174, 15, 163bitr4i 285 1
 Colors of variables: wff setvar class Syntax hints:   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904   wss 3390  cpw 3942   cxp 4837  cima 4842  cc 9555  chil 26653   cva 26654   csm 26655  c0v 26658  csh 26662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-hilex 26733 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-sh 26941 This theorem is referenced by:  issh2  26943  shss  26944  sh0  26950
 Copyright terms: Public domain W3C validator