Theorem issh 28065

Description: Subspace H 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	|- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )

Proof of Theorem issh

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-hilex 27856	. . . 4 |- ~H e. _V
2	1	elpw2 4828	. . 3 |- ( H e. ~P ~H <-> H C_ ~H )
3		3anass 1042	. . 3 |- ( ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) <-> ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )
4	2, 3	anbi12i 733	. 2 |- ( ( H e. ~P ~H /\ ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) <-> ( H C_ ~H /\ ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) )
5		eleq2 2690	. . . 4 |- ( h = H -> ( 0h e. h <-> 0h e. H ) )
6		id 22	. . . . . . 7 |- ( h = H -> h = H )
7	6	sqxpeqd 5141	. . . . . 6 |- ( h = H -> ( h X. h ) = ( H X. H ) )
8	7	imaeq2d 5466	. . . . 5 |- ( h = H -> ( +h " ( h X. h ) ) = ( +h " ( H X. H ) ) )
9	8, 6	sseq12d 3634	. . . 4 |- ( h = H -> ( ( +h " ( h X. h ) ) C_ h <-> ( +h " ( H X. H ) ) C_ H ) )
10		xpeq2 5129	. . . . . 6 |- ( h = H -> ( CC X. h ) = ( CC X. H ) )
11	10	imaeq2d 5466	. . . . 5 |- ( h = H -> ( .h " ( CC X. h ) ) = ( .h " ( CC X. H ) ) )
12	11, 6	sseq12d 3634	. . . 4 |- ( h = H -> ( ( .h " ( CC X. h ) ) C_ h <-> ( .h " ( CC X. H ) ) C_ H ) )
13	5, 9, 12	3anbi123d 1399	. . 3 |- ( h = H -> ( ( 0h e. h /\ ( +h " ( h X. h ) ) C_ h /\ ( .h " ( CC X. h ) ) C_ h ) <-> ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )
14		df-sh 28064	. . 3 |- SH = { h e. ~P ~H | ( 0h e. h /\ ( +h " ( h X. h ) ) C_ h /\ ( .h " ( CC X. h ) ) C_ h ) }
15	13, 14	elrab2 3366	. 2 |- ( H e. SH <-> ( H e. ~P ~H /\ ( 0h e. H /\ ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )
16		anass 681	. 2 |- ( ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) <-> ( H C_ ~H /\ ( 0h e. H /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) ) )
17	4, 15, 16	3bitr4i 292	1 |- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   X. cxp 5112   "cima 5117   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-hilex 27856

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-sh 28064

This theorem is referenced by:  issh2  28066  shss  28067  sh0  28073