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Theorem hhsssh 24621
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 24618 . . 3  |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U )
)
4 shss 24563 . . 3  |-  ( H  e.  SH  ->  H  C_ 
~H )
53, 4jca 532 . 2  |-  ( H  e.  SH  ->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) )
6 eleq1 2498 . . 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 2438 . . . 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 4849 . . . . . . . . . . . . 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 4850 . . . . . . . . . . . . 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 2470 . . . . . . . . . . . 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 5105 . . . . . . . . . . 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 4850 . . . . . . . . . . . 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 5105 . . . . . . . . . . 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 4062 . . . . . . . . . 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 5100 . . . . . . . . . 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 4062 . . . . . . . . 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 2482 . . . . . . . 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 2504 . . . . . . 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 3372 . . . . . . 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 710 . . . . . 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 4849 . . . . . . . . . . . 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 4850 . . . . . . . . . . . 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 2470 . . . . . . . . . . 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 5105 . . . . . . . . . 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 4850 . . . . . . . . . . 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 5105 . . . . . . . . . 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 4062 . . . . . . . . 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 5100 . . . . . . . . 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 4062 . . . . . . . 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 2504 . . . . . . 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 3372 . . . . . . 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 710 . . . . . 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 24353 . . . . . . . . . . . 12  |-  +h  :
( ~H  X.  ~H )
--> ~H
34 ffn 5554 . . . . . . . . . . . 12  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  +h  Fn  ( ~H  X.  ~H )
)
35 fnresdm 5515 . . . . . . . . . . . 12  |-  (  +h  Fn  ( ~H  X.  ~H )  ->  (  +h  |`  ( ~H  X.  ~H ) )  =  +h  )
3633, 34, 35mp2b 10 . . . . . . . . . . 11  |-  (  +h  |`  ( ~H  X.  ~H ) )  =  +h
37 ax-hfvmul 24358 . . . . . . . . . . . 12  |-  .h  :
( CC  X.  ~H )
--> ~H
38 ffn 5554 . . . . . . . . . . . 12  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
39 fnresdm 5515 . . . . . . . . . . . 12  |-  (  .h  Fn  ( CC  X.  ~H )  ->  (  .h  |`  ( CC  X.  ~H ) )  =  .h  )
4037, 38, 39mp2b 10 . . . . . . . . . . 11  |-  (  .h  |`  ( CC  X.  ~H ) )  =  .h
4136, 40opeq12i 4059 . . . . . . . . . 10  |-  <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >.  =  <.  +h  ,  .h  >.
42 normf 24476 . . . . . . . . . . 11  |-  normh : ~H --> RR
43 ffn 5554 . . . . . . . . . . 11  |-  ( normh : ~H --> RR  ->  normh  Fn  ~H )
44 fnresdm 5515 . . . . . . . . . . 11  |-  ( normh  Fn 
~H  ->  ( normh  |`  ~H )  =  normh )
4542, 43, 44mp2b 10 . . . . . . . . . 10  |-  ( normh  |`  ~H )  =  normh
4641, 45opeq12i 4059 . . . . . . . . 9  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  <. <.  +h  ,  .h  >. ,  normh >.
4746, 1eqtr4i 2461 . . . . . . . 8  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  U
481hhnv 24518 . . . . . . . . 9  |-  U  e.  NrmCVec
49 eqid 2438 . . . . . . . . . 10  |-  ( SubSp `  U )  =  (
SubSp `  U )
5049sspid 24074 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  U  e.  (
SubSp `  U ) )
5148, 50ax-mp 5 . . . . . . . 8  |-  U  e.  ( SubSp `  U )
5247, 51eqeltri 2508 . . . . . . 7  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U
)
53 ssid 3370 . . . . . . 7  |-  ~H  C_  ~H
5452, 53pm3.2i 455 . . . . . 6  |-  ( <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) )
>. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U )  /\  ~H  C_  ~H )
5520, 32, 54elimhyp 3843 . . . . 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 458 . . . 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 462 . . . 4  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  C_ 
~H
581, 7, 56, 57hhshsslem2 24620 . . 3  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  e.  SH
596, 58dedth 3836 . 2  |-  ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H )  ->  H  e.  SH )
605, 59impbii 188 1  |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_  ~H )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3323   ifcif 3786   <.cop 3878    X. cxp 4833    |` cres 4837    Fn wfn 5408   -->wf 5409   ` cfv 5413   CCcc 9272   RRcr 9273   NrmCVeccnv 23913   SubSpcss 24070   ~Hchil 24272    +h cva 24273    .h csm 24274   normhcno 24276   SHcsh 24281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354  ax-hilex 24352  ax-hfvadd 24353  ax-hvcom 24354  ax-hvass 24355  ax-hv0cl 24356  ax-hvaddid 24357  ax-hfvmul 24358  ax-hvmulid 24359  ax-hvmulass 24360  ax-hvdistr1 24361  ax-hvdistr2 24362  ax-hvmul0 24363  ax-hfi 24432  ax-his1 24435  ax-his2 24436  ax-his3 24437  ax-his4 24438
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-icc 11299  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-topgen 14374  df-psmet 17784  df-xmet 17785  df-met 17786  df-bl 17787  df-mopn 17788  df-top 18478  df-bases 18480  df-topon 18481  df-lm 18808  df-haus 18894  df-grpo 23629  df-gid 23630  df-ginv 23631  df-gdiv 23632  df-ablo 23720  df-subgo 23740  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-vs 23928  df-nmcv 23929  df-ims 23930  df-ssp 24071  df-hnorm 24321  df-hba 24322  df-hvsub 24324  df-hlim 24325  df-sh 24560  df-ch 24575  df-ch0 24607
This theorem is referenced by:  hhsssh2  24622
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