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Theorem hhsssh 21862
Description: The predicate " H is a subspace of Hilbert space." (Contributed by NM, 25-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
hhsst.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
hhsst.2  |-  W  = 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
Assertion
Ref Expression
hhsssh  |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_  ~H )
)

Proof of Theorem hhsssh
StepHypRef Expression
1 hhsst.1 . . . 4  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
2 hhsst.2 . . . 4  |-  W  = 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
31, 2hhsst 21859 . . 3  |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U )
)
4 shss 21805 . . 3  |-  ( H  e.  SH  ->  H  C_ 
~H )
53, 4jca 518 . 2  |-  ( H  e.  SH  ->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) )
6 eleq1 2356 . . 3  |-  ( 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 2296 . . . 4  |-  <. <. (  +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 4719 . . . . . . . . . . . . 13  |-  ( 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 4720 . . . . . . . . . . . . 13  |-  ( 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 )
) )
108, 9eqtrd 2328 . . . . . . . . . . . 12  |-  ( 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 )
) )
1110reseq2d 4971 . . . . . . . . . . 11  |-  ( 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 4720 . . . . . . . . . . . 12  |-  ( 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 )
) )
1312reseq2d 4971 . . . . . . . . . . 11  |-  ( 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 ) ) ) )
1411, 13opeq12d 3820 . . . . . . . . . 10  |-  ( 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 4966 . . . . . . . . . 10  |-  ( H  =  if ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) ,  H ,  ~H )  ->  ( normh 
|`  H )  =  ( normh  |`  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )
) )
1614, 15opeq12d 3820 . . . . . . . . 9  |-  ( 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 )
) >. )
172, 16syl5eq 2340 . . . . . . . 8  |-  ( 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 )
) >. )
1817eleq1d 2362 . . . . . . 7  |-  ( 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 3212 . . . . . . 7  |-  ( 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 ) )
2018, 19anbi12d 691 . . . . . 6  |-  ( 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 4719 . . . . . . . . . . . 12  |-  ( ~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 4720 . . . . . . . . . . . 12  |-  ( ~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 )
) )
2321, 22eqtrd 2328 . . . . . . . . . . 11  |-  ( ~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 )
) )
2423reseq2d 4971 . . . . . . . . . 10  |-  ( ~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 4720 . . . . . . . . . . 11  |-  ( ~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 )
) )
2625reseq2d 4971 . . . . . . . . . 10  |-  ( ~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 ) ) ) )
2724, 26opeq12d 3820 . . . . . . . . 9  |-  ( ~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 4966 . . . . . . . . 9  |-  ( ~H  =  if ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) ,  H ,  ~H )  ->  ( normh 
|`  ~H )  =  (
normh  |`  if ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) ,  H ,  ~H ) ) )
2927, 28opeq12d 3820 . . . . . . . 8  |-  ( ~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 )
) >. )
3029eleq1d 2362 . . . . . . 7  |-  ( ~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 3212 . . . . . . 7  |-  ( ~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 ) )
3230, 31anbi12d 691 . . . . . 6  |-  ( ~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 ) ) )
33 ax-hfvadd 21596 . . . . . . . . . . . 12  |-  +h  :
( ~H  X.  ~H )
--> ~H
34 ffn 5405 . . . . . . . . . . . 12  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  +h  Fn  ( ~H  X.  ~H )
)
35 fnresdm 5369 . . . . . . . . . . . 12  |-  (  +h  Fn  ( ~H  X.  ~H )  ->  (  +h  |`  ( ~H  X.  ~H ) )  =  +h  )
3633, 34, 35mp2b 9 . . . . . . . . . . 11  |-  (  +h  |`  ( ~H  X.  ~H ) )  =  +h
37 ax-hfvmul 21601 . . . . . . . . . . . 12  |-  .h  :
( CC  X.  ~H )
--> ~H
38 ffn 5405 . . . . . . . . . . . 12  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
39 fnresdm 5369 . . . . . . . . . . . 12  |-  (  .h  Fn  ( CC  X.  ~H )  ->  (  .h  |`  ( CC  X.  ~H ) )  =  .h  )
4037, 38, 39mp2b 9 . . . . . . . . . . 11  |-  (  .h  |`  ( CC  X.  ~H ) )  =  .h
4136, 40opeq12i 3817 . . . . . . . . . 10  |-  <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >.  =  <.  +h  ,  .h  >.
42 normf 21718 . . . . . . . . . . 11  |-  normh : ~H --> RR
43 ffn 5405 . . . . . . . . . . 11  |-  ( normh : ~H --> RR  ->  normh  Fn  ~H )
44 fnresdm 5369 . . . . . . . . . . 11  |-  ( normh  Fn 
~H  ->  ( normh  |`  ~H )  =  normh )
4542, 43, 44mp2b 9 . . . . . . . . . 10  |-  ( normh  |`  ~H )  =  normh
4641, 45opeq12i 3817 . . . . . . . . 9  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  <. <.  +h  ,  .h  >. ,  normh >.
4746, 1eqtr4i 2319 . . . . . . . 8  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  U
481hhnv 21760 . . . . . . . . 9  |-  U  e.  NrmCVec
49 eqid 2296 . . . . . . . . . 10  |-  ( SubSp `  U )  =  (
SubSp `  U )
5049sspid 21317 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  U  e.  (
SubSp `  U ) )
5148, 50ax-mp 8 . . . . . . . 8  |-  U  e.  ( SubSp `  U )
5247, 51eqeltri 2366 . . . . . . 7  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U
)
53 ssid 3210 . . . . . . 7  |-  ~H  C_  ~H
5452, 53pm3.2i 441 . . . . . 6  |-  ( <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) )
>. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U )  /\  ~H  C_  ~H )
5520, 32, 54elimhyp 3626 . . . . 5  |-  ( <. <. (  +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 )
5655simpli 444 . . . 4  |-  <. <. (  +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 )
5755simpri 448 . . . 4  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  C_ 
~H
581, 7, 56, 57hhshsslem2 21861 . . 3  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  e.  SH
596, 58dedth 3619 . 2  |-  ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H )  ->  H  e.  SH )
605, 59impbii 180 1  |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_  ~H )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    C_ wss 3165   ifcif 3578   <.cop 3656    X. cxp 4703    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271   CCcc 8751   RRcr 8752   NrmCVeccnv 21156   SubSpcss 21313   ~Hchil 21515    +h cva 21516    .h csm 21517   normhcno 21519   SHcsh 21524
This theorem is referenced by:  hhsssh2  21863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-icc 10679  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-lm 16975  df-haus 17059  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-ssp 21314  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-sh 21802  df-ch 21817  df-ch0 21848
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