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Theorem hhsssh 26920
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 26917 . . 3  |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U )
)
4 shss 26863 . . 3  |-  ( H  e.  SH  ->  H  C_ 
~H )
53, 4jca 535 . 2  |-  ( H  e.  SH  ->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) )
6 eleq1 2517 . . 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 2451 . . . 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 4848 . . . . . . . . . . . . 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 4849 . . . . . . . . . . . . 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 2485 . . . . . . . . . . . 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 4849 . . . . . . . . . . . 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 4174 . . . . . . . . . 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 4174 . . . . . . . . 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 2497 . . . . . . . 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 2513 . . . . . . 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 3453 . . . . . . 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 717 . . . . . 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 4848 . . . . . . . . . . . 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 4849 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 4849 . . . . . . . . . . 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 4174 . . . . . . . . 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 4174 . . . . . . . 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 2513 . . . . . . 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 3453 . . . . . . 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 717 . . . . . 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 26653 . . . . . . . . . . . 12  |-  +h  :
( ~H  X.  ~H )
--> ~H
34 ffn 5728 . . . . . . . . . . . 12  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  +h  Fn  ( ~H  X.  ~H )
)
35 fnresdm 5685 . . . . . . . . . . . 12  |-  (  +h  Fn  ( ~H  X.  ~H )  ->  (  +h  |`  ( ~H  X.  ~H ) )  =  +h  )
3633, 34, 35mp2b 10 . . . . . . . . . . 11  |-  (  +h  |`  ( ~H  X.  ~H ) )  =  +h
37 ax-hfvmul 26658 . . . . . . . . . . . 12  |-  .h  :
( CC  X.  ~H )
--> ~H
38 ffn 5728 . . . . . . . . . . . 12  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
39 fnresdm 5685 . . . . . . . . . . . 12  |-  (  .h  Fn  ( CC  X.  ~H )  ->  (  .h  |`  ( CC  X.  ~H ) )  =  .h  )
4037, 38, 39mp2b 10 . . . . . . . . . . 11  |-  (  .h  |`  ( CC  X.  ~H ) )  =  .h
4136, 40opeq12i 4171 . . . . . . . . . 10  |-  <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >.  =  <.  +h  ,  .h  >.
42 normf 26776 . . . . . . . . . . 11  |-  normh : ~H --> RR
43 ffn 5728 . . . . . . . . . . 11  |-  ( normh : ~H --> RR  ->  normh  Fn  ~H )
44 fnresdm 5685 . . . . . . . . . . 11  |-  ( normh  Fn 
~H  ->  ( normh  |`  ~H )  =  normh )
4542, 43, 44mp2b 10 . . . . . . . . . 10  |-  ( normh  |`  ~H )  =  normh
4641, 45opeq12i 4171 . . . . . . . . 9  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  <. <.  +h  ,  .h  >. ,  normh >.
4746, 1eqtr4i 2476 . . . . . . . 8  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  U
481hhnv 26818 . . . . . . . . 9  |-  U  e.  NrmCVec
49 eqid 2451 . . . . . . . . . 10  |-  ( SubSp `  U )  =  (
SubSp `  U )
5049sspid 26364 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  U  e.  (
SubSp `  U ) )
5148, 50ax-mp 5 . . . . . . . 8  |-  U  e.  ( SubSp `  U )
5247, 51eqeltri 2525 . . . . . . 7  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U
)
53 ssid 3451 . . . . . . 7  |-  ~H  C_  ~H
5452, 53pm3.2i 457 . . . . . 6  |-  ( <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) )
>. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U )  /\  ~H  C_  ~H )
5520, 32, 54elimhyp 3939 . . . . 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 460 . . . 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 464 . . . 4  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  C_ 
~H
581, 7, 56, 57hhshsslem2 26919 . . 3  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  e.  SH
596, 58dedth 3932 . 2  |-  ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H )  ->  H  e.  SH )
605, 59impbii 191 1  |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_  ~H )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887    C_ wss 3404   ifcif 3881   <.cop 3974    X. cxp 4832    |` cres 4836    Fn wfn 5577   -->wf 5578   ` cfv 5582   CCcc 9537   RRcr 9538   NrmCVeccnv 26203   SubSpcss 26360   ~Hchil 26572    +h cva 26573    .h csm 26574   normhcno 26576   SHcsh 26581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619  ax-hilex 26652  ax-hfvadd 26653  ax-hvcom 26654  ax-hvass 26655  ax-hv0cl 26656  ax-hvaddid 26657  ax-hfvmul 26658  ax-hvmulid 26659  ax-hvmulass 26660  ax-hvdistr1 26661  ax-hvdistr2 26662  ax-hvmul0 26663  ax-hfi 26732  ax-his1 26735  ax-his2 26736  ax-his3 26737  ax-his4 26738
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-pm 7475  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-icc 11642  df-seq 12214  df-exp 12273  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-topgen 15342  df-psmet 18962  df-xmet 18963  df-met 18964  df-bl 18965  df-mopn 18966  df-top 19921  df-bases 19922  df-topon 19923  df-lm 20245  df-haus 20331  df-grpo 25919  df-gid 25920  df-ginv 25921  df-gdiv 25922  df-ablo 26010  df-subgo 26030  df-vc 26165  df-nv 26211  df-va 26214  df-ba 26215  df-sm 26216  df-0v 26217  df-vs 26218  df-nmcv 26219  df-ims 26220  df-ssp 26361  df-hnorm 26621  df-hba 26622  df-hvsub 26624  df-hlim 26625  df-sh 26860  df-ch 26874  df-ch0 26906
This theorem is referenced by:  hhsssh2  26921
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