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Theorem hhsssh 22619
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 22616 . . 3  |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U )
)
4 shss 22562 . . 3  |-  ( H  e.  SH  ->  H  C_ 
~H )
53, 4jca 519 . 2  |-  ( H  e.  SH  ->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) )
6 eleq1 2449 . . 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 2389 . . . 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 4834 . . . . . . . . . . . . 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 4835 . . . . . . . . . . . . 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 2421 . . . . . . . . . . . 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 5088 . . . . . . . . . . 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 4835 . . . . . . . . . . . 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 5088 . . . . . . . . . . 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 3936 . . . . . . . . . 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 5083 . . . . . . . . . 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 3936 . . . . . . . . 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 2433 . . . . . . . 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 2455 . . . . . . 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 3314 . . . . . . 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 692 . . . . . 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 4834 . . . . . . . . . . . 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 4835 . . . . . . . . . . . 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 2421 . . . . . . . . . . 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 5088 . . . . . . . . . 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 4835 . . . . . . . . . . 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 5088 . . . . . . . . . 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 3936 . . . . . . . . 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 5083 . . . . . . . . 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 3936 . . . . . . . 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 2455 . . . . . . 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 3314 . . . . . . 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 692 . . . . . 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 22353 . . . . . . . . . . . 12  |-  +h  :
( ~H  X.  ~H )
--> ~H
34 ffn 5533 . . . . . . . . . . . 12  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  +h  Fn  ( ~H  X.  ~H )
)
35 fnresdm 5496 . . . . . . . . . . . 12  |-  (  +h  Fn  ( ~H  X.  ~H )  ->  (  +h  |`  ( ~H  X.  ~H ) )  =  +h  )
3633, 34, 35mp2b 10 . . . . . . . . . . 11  |-  (  +h  |`  ( ~H  X.  ~H ) )  =  +h
37 ax-hfvmul 22358 . . . . . . . . . . . 12  |-  .h  :
( CC  X.  ~H )
--> ~H
38 ffn 5533 . . . . . . . . . . . 12  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
39 fnresdm 5496 . . . . . . . . . . . 12  |-  (  .h  Fn  ( CC  X.  ~H )  ->  (  .h  |`  ( CC  X.  ~H ) )  =  .h  )
4037, 38, 39mp2b 10 . . . . . . . . . . 11  |-  (  .h  |`  ( CC  X.  ~H ) )  =  .h
4136, 40opeq12i 3933 . . . . . . . . . 10  |-  <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >.  =  <.  +h  ,  .h  >.
42 normf 22475 . . . . . . . . . . 11  |-  normh : ~H --> RR
43 ffn 5533 . . . . . . . . . . 11  |-  ( normh : ~H --> RR  ->  normh  Fn  ~H )
44 fnresdm 5496 . . . . . . . . . . 11  |-  ( normh  Fn 
~H  ->  ( normh  |`  ~H )  =  normh )
4542, 43, 44mp2b 10 . . . . . . . . . 10  |-  ( normh  |`  ~H )  =  normh
4641, 45opeq12i 3933 . . . . . . . . 9  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  <. <.  +h  ,  .h  >. ,  normh >.
4746, 1eqtr4i 2412 . . . . . . . 8  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  U
481hhnv 22517 . . . . . . . . 9  |-  U  e.  NrmCVec
49 eqid 2389 . . . . . . . . . 10  |-  ( SubSp `  U )  =  (
SubSp `  U )
5049sspid 22074 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  U  e.  (
SubSp `  U ) )
5148, 50ax-mp 8 . . . . . . . 8  |-  U  e.  ( SubSp `  U )
5247, 51eqeltri 2459 . . . . . . 7  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U
)
53 ssid 3312 . . . . . . 7  |-  ~H  C_  ~H
5452, 53pm3.2i 442 . . . . . 6  |-  ( <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) )
>. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U )  /\  ~H  C_  ~H )
5520, 32, 54elimhyp 3732 . . . . 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 445 . . . 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 449 . . . 4  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  C_ 
~H
581, 7, 56, 57hhshsslem2 22618 . . 3  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  e.  SH
596, 58dedth 3725 . 2  |-  ( ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H )  ->  H  e.  SH )
605, 59impbii 181 1  |-  ( H  e.  SH  <->  ( W  e.  ( SubSp `  U )  /\  H  C_  ~H )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265   ifcif 3684   <.cop 3762    X. cxp 4818    |` cres 4822    Fn wfn 5391   -->wf 5392   ` cfv 5396   CCcc 8923   RRcr 8924   NrmCVeccnv 21913   SubSpcss 22070   ~Hchil 22272    +h cva 22273    .h csm 22274   normhcno 22276   SHcsh 22281
This theorem is referenced by:  hhsssh2  22620
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005  ax-hilex 22352  ax-hfvadd 22353  ax-hvcom 22354  ax-hvass 22355  ax-hv0cl 22356  ax-hvaddid 22357  ax-hfvmul 22358  ax-hvmulid 22359  ax-hvmulass 22360  ax-hvdistr1 22361  ax-hvdistr2 22362  ax-hvmul0 22363  ax-hfi 22431  ax-his1 22434  ax-his2 22435  ax-his3 22436  ax-his4 22437
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-map 6958  df-pm 6959  df-en 7048  df-dom 7049  df-sdom 7050  df-sup 7383  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-n0 10156  df-z 10217  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-icc 10857  df-seq 11253  df-exp 11312  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-topgen 13596  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-top 16888  df-bases 16890  df-topon 16891  df-lm 17217  df-haus 17303  df-grpo 21629  df-gid 21630  df-ginv 21631  df-gdiv 21632  df-ablo 21720  df-subgo 21740  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-vs 21928  df-nmcv 21929  df-ims 21930  df-ssp 22071  df-hnorm 22321  df-hba 22322  df-hvsub 22324  df-hlim 22325  df-sh 22559  df-ch 22574  df-ch0 22605
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