# Metamath Proof Explorer

## Theorem hhsssh

Description: The predicate " H is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008) (New usage is discouraged.)

Ref Expression
Hypotheses hhsst.1
`|- U = <. <. +h , .h >. , normh >.`
hhsst.2
`|- W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.`
Assertion hhsssh
`|- ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )`

### Proof

Step Hyp Ref Expression
1 hhsst.1
` |-  U = <. <. +h , .h >. , normh >.`
2 hhsst.2
` |-  W = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >.`
3 1 2 hhsst
` |-  ( H e. SH -> W e. ( SubSp ` U ) )`
4 shss
` |-  ( H e. SH -> H C_ ~H )`
5 3 4 jca
` |-  ( H e. SH -> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )`
6 eleq1
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H e. SH <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH ) )`
7 eqid
` |-  <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >.`
8 xpeq1
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. H ) )`
9 xpeq2
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
10 8 9 eqtrd
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H X. H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
11 10 reseq2d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( +h |` ( H X. H ) ) = ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )`
12 xpeq2
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( CC X. H ) = ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
13 12 reseq2d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( .h |` ( CC X. H ) ) = ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )`
14 11 13 opeq12d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. = <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. )`
15 reseq2
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( normh |` H ) = ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
16 14 15 opeq12d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )`
17 2 16 syl5eq
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> W = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )`
18 17 eleq1d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( W e. ( SubSp ` U ) <-> <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) ) )`
19 sseq1
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( H C_ ~H <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) )`
20 18 19 anbi12d
` |-  ( H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) <-> ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) ) )`
21 xpeq1
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. ~H ) )`
22 xpeq2
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
23 21 22 eqtrd
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H X. ~H ) = ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
24 23 reseq2d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( +h |` ( ~H X. ~H ) ) = ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )`
25 xpeq2
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( CC X. ~H ) = ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
26 25 reseq2d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( .h |` ( CC X. ~H ) ) = ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) )`
27 24 26 opeq12d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. )`
28 reseq2
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( normh |` ~H ) = ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) )`
29 27 28 opeq12d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. )`
30 29 eleq1d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) <-> <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) ) )`
31 sseq1
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ~H C_ ~H <-> if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) )`
32 30 31 anbi12d
` |-  ( ~H = if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) -> ( ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H ) <-> ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H ) ) )`
` |-  +h : ( ~H X. ~H ) --> ~H`
34 ffn
` |-  ( +h : ( ~H X. ~H ) --> ~H -> +h Fn ( ~H X. ~H ) )`
35 fnresdm
` |-  ( +h Fn ( ~H X. ~H ) -> ( +h |` ( ~H X. ~H ) ) = +h )`
36 33 34 35 mp2b
` |-  ( +h |` ( ~H X. ~H ) ) = +h`
37 ax-hfvmul
` |-  .h : ( CC X. ~H ) --> ~H`
38 ffn
` |-  ( .h : ( CC X. ~H ) --> ~H -> .h Fn ( CC X. ~H ) )`
39 fnresdm
` |-  ( .h Fn ( CC X. ~H ) -> ( .h |` ( CC X. ~H ) ) = .h )`
40 37 38 39 mp2b
` |-  ( .h |` ( CC X. ~H ) ) = .h`
41 36 40 opeq12i
` |-  <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. = <. +h , .h >.`
42 normf
` |-  normh : ~H --> RR`
43 ffn
` |-  ( normh : ~H --> RR -> normh Fn ~H )`
44 fnresdm
` |-  ( normh Fn ~H -> ( normh |` ~H ) = normh )`
45 42 43 44 mp2b
` |-  ( normh |` ~H ) = normh`
46 41 45 opeq12i
` |-  <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = <. <. +h , .h >. , normh >.`
47 46 1 eqtr4i
` |-  <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. = U`
48 1 hhnv
` |-  U e. NrmCVec`
49 eqid
` |-  ( SubSp ` U ) = ( SubSp ` U )`
50 49 sspid
` |-  ( U e. NrmCVec -> U e. ( SubSp ` U ) )`
51 48 50 ax-mp
` |-  U e. ( SubSp ` U )`
52 47 51 eqeltri
` |-  <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U )`
53 ssid
` |-  ~H C_ ~H`
54 52 53 pm3.2i
` |-  ( <. <. ( +h |` ( ~H X. ~H ) ) , ( .h |` ( CC X. ~H ) ) >. , ( normh |` ~H ) >. e. ( SubSp ` U ) /\ ~H C_ ~H )`
55 20 32 54 elimhyp
` |-  ( <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U ) /\ if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H )`
56 55 simpli
` |-  <. <. ( +h |` ( if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) , ( .h |` ( CC X. if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) ) >. , ( normh |` if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) ) >. e. ( SubSp ` U )`
57 55 simpri
` |-  if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) C_ ~H`
58 1 7 56 57 hhshsslem2
` |-  if ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) , H , ~H ) e. SH`
59 6 58 dedth
` |-  ( ( W e. ( SubSp ` U ) /\ H C_ ~H ) -> H e. SH )`
60 5 59 impbii
` |-  ( H e. SH <-> ( W e. ( SubSp ` U ) /\ H C_ ~H ) )`