Theorem issh2 28066

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. Definition of [Beran] p. 95. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
issh2	|- ( 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 ) ) )

Distinct variable group:   x, y, H

Proof of Theorem issh2

Step	Hyp	Ref	Expression
1		issh 28065	. 2 |- ( H e. SH <-> ( ( H C_ ~H /\ 0h e. H ) /\ ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ H ) ) )
2		ax-hfvadd 27857	. . . . . . 7 |- +h : ( ~H X. ~H ) --> ~H
3		ffun 6048	. . . . . . 7 |- ( +h : ( ~H X. ~H ) --> ~H -> Fun +h )
4	2, 3	ax-mp 5	. . . . . 6 |- Fun +h
5		xpss12 5225	. . . . . . . 8 |- ( ( H C_ ~H /\ H C_ ~H ) -> ( H X. H ) C_ ( ~H X. ~H ) )
6	5	anidms 677	. . . . . . 7 |- ( H C_ ~H -> ( H X. H ) C_ ( ~H X. ~H ) )
7	2	fdmi 6052	. . . . . . 7 |- dom +h = ( ~H X. ~H )
8	6, 7	syl6sseqr 3652	. . . . . 6 |- ( H C_ ~H -> ( H X. H ) C_ dom +h )
9		funimassov 6811	. . . . . 6 |- ( ( Fun +h /\ ( H X. H ) C_ dom +h ) -> ( ( +h " ( H X. H ) ) C_ H <-> A. x e. H A. y e. H ( x +h y ) e. H ) )
10	4, 8, 9	sylancr 695	. . . . 5 |- ( H C_ ~H -> ( ( +h " ( H X. H ) ) C_ H <-> A. x e. H A. y e. H ( x +h y ) e. H ) )
11		ax-hfvmul 27862	. . . . . . 7 |- .h : ( CC X. ~H ) --> ~H
12		ffun 6048	. . . . . . 7 |- ( .h : ( CC X. ~H ) --> ~H -> Fun .h )
13	11, 12	ax-mp 5	. . . . . 6 |- Fun .h
14		xpss2 5229	. . . . . . 7 |- ( H C_ ~H -> ( CC X. H ) C_ ( CC X. ~H ) )
15	11	fdmi 6052	. . . . . . 7 |- dom .h = ( CC X. ~H )
16	14, 15	syl6sseqr 3652	. . . . . 6 |- ( H C_ ~H -> ( CC X. H ) C_ dom .h )
17		funimassov 6811	. . . . . 6 |- ( ( Fun .h /\ ( CC X. H ) C_ dom .h ) -> ( ( .h " ( CC X. H ) ) C_ H <-> A. x e. CC A. y e. H ( x .h y ) e. H ) )
18	13, 16, 17	sylancr 695	. . . . 5 |- ( H C_ ~H -> ( ( .h " ( CC X. H ) ) C_ H <-> A. x e. CC A. y e. H ( x .h y ) e. H ) )
19	10, 18	anbi12d 747	. . . 4 |- ( H C_ ~H -> ( ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ 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 ) ) )
20	19	adantr 481	. . 3 |- ( ( H C_ ~H /\ 0h e. H ) -> ( ( ( +h " ( H X. H ) ) C_ H /\ ( .h " ( CC X. H ) ) C_ 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 ) ) )
21	20	pm5.32i 669	. 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 ) /\ ( 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 ) ) )
22	1, 21	bitri 264	1 |- ( 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 ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   X. cxp 5112   dom cdm 5114   "cima 5117   Fun wfun 5882   -->wf 5884  (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:  shaddcl  28074  shmulcl  28075  issh3  28076  helch  28100  hsn0elch  28105  hhshsslem2  28125  ocsh  28142  shscli  28176  shintcli  28188  imaelshi  28917