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Theorem hhsssh 24842
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 24839 . . 3  |-  ( H  e.  SH  ->  W  e.  ( SubSp `  U )
)
4 shss 24784 . . 3  |-  ( H  e.  SH  ->  H  C_ 
~H )
53, 4jca 532 . 2  |-  ( H  e.  SH  ->  ( W  e.  ( SubSp `  U )  /\  H  C_ 
~H ) )
6 eleq1 2526 . . 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 2454 . . . 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 4965 . . . . . . . . . . . . 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 4966 . . . . . . . . . . . . 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 2495 . . . . . . . . . . . 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 5221 . . . . . . . . . . 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 4966 . . . . . . . . . . . 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 5221 . . . . . . . . . . 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 4178 . . . . . . . . . 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 5216 . . . . . . . . . 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 4178 . . . . . . . . 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 2507 . . . . . . . 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 2523 . . . . . . 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 3488 . . . . . . 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 4965 . . . . . . . . . . . 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 4966 . . . . . . . . . . . 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 2495 . . . . . . . . . . 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 5221 . . . . . . . . . 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 4966 . . . . . . . . . . 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 5221 . . . . . . . . . 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 4178 . . . . . . . . 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 5216 . . . . . . . . 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 4178 . . . . . . . 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 2523 . . . . . . 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 3488 . . . . . . 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 24574 . . . . . . . . . . . 12  |-  +h  :
( ~H  X.  ~H )
--> ~H
34 ffn 5670 . . . . . . . . . . . 12  |-  (  +h  : ( ~H  X.  ~H ) --> ~H  ->  +h  Fn  ( ~H  X.  ~H )
)
35 fnresdm 5631 . . . . . . . . . . . 12  |-  (  +h  Fn  ( ~H  X.  ~H )  ->  (  +h  |`  ( ~H  X.  ~H ) )  =  +h  )
3633, 34, 35mp2b 10 . . . . . . . . . . 11  |-  (  +h  |`  ( ~H  X.  ~H ) )  =  +h
37 ax-hfvmul 24579 . . . . . . . . . . . 12  |-  .h  :
( CC  X.  ~H )
--> ~H
38 ffn 5670 . . . . . . . . . . . 12  |-  (  .h  : ( CC  X.  ~H ) --> ~H  ->  .h  Fn  ( CC  X.  ~H )
)
39 fnresdm 5631 . . . . . . . . . . . 12  |-  (  .h  Fn  ( CC  X.  ~H )  ->  (  .h  |`  ( CC  X.  ~H ) )  =  .h  )
4037, 38, 39mp2b 10 . . . . . . . . . . 11  |-  (  .h  |`  ( CC  X.  ~H ) )  =  .h
4136, 40opeq12i 4175 . . . . . . . . . 10  |-  <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >.  =  <.  +h  ,  .h  >.
42 normf 24697 . . . . . . . . . . 11  |-  normh : ~H --> RR
43 ffn 5670 . . . . . . . . . . 11  |-  ( normh : ~H --> RR  ->  normh  Fn  ~H )
44 fnresdm 5631 . . . . . . . . . . 11  |-  ( normh  Fn 
~H  ->  ( normh  |`  ~H )  =  normh )
4542, 43, 44mp2b 10 . . . . . . . . . 10  |-  ( normh  |`  ~H )  =  normh
4641, 45opeq12i 4175 . . . . . . . . 9  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  <. <.  +h  ,  .h  >. ,  normh >.
4746, 1eqtr4i 2486 . . . . . . . 8  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  =  U
481hhnv 24739 . . . . . . . . 9  |-  U  e.  NrmCVec
49 eqid 2454 . . . . . . . . . 10  |-  ( SubSp `  U )  =  (
SubSp `  U )
5049sspid 24295 . . . . . . . . 9  |-  ( U  e.  NrmCVec  ->  U  e.  (
SubSp `  U ) )
5148, 50ax-mp 5 . . . . . . . 8  |-  U  e.  ( SubSp `  U )
5247, 51eqeltri 2538 . . . . . . 7  |-  <. <. (  +h  |`  ( ~H  X.  ~H ) ) ,  (  .h  |`  ( CC  X.  ~H ) ) >. ,  ( normh  |`  ~H ) >.  e.  ( SubSp `  U
)
53 ssid 3486 . . . . . . 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 3959 . . . . 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 24841 . . 3  |-  if ( ( W  e.  (
SubSp `  U )  /\  H  C_  ~H ) ,  H ,  ~H )  e.  SH
596, 58dedth 3952 . 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 1370    e. wcel 1758    C_ wss 3439   ifcif 3902   <.cop 3994    X. cxp 4949    |` cres 4953    Fn wfn 5524   -->wf 5525   ` cfv 5529   CCcc 9394   RRcr 9395   NrmCVeccnv 24134   SubSpcss 24291   ~Hchil 24493    +h cva 24494    .h csm 24495   normhcno 24497   SHcsh 24502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474  ax-addf 9475  ax-mulf 9476  ax-hilex 24573  ax-hfvadd 24574  ax-hvcom 24575  ax-hvass 24576  ax-hv0cl 24577  ax-hvaddid 24578  ax-hfvmul 24579  ax-hvmulid 24580  ax-hvmulass 24581  ax-hvdistr1 24582  ax-hvdistr2 24583  ax-hvmul0 24584  ax-hfi 24653  ax-his1 24656  ax-his2 24657  ax-his3 24658  ax-his4 24659
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-sup 7805  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-n0 10694  df-z 10761  df-uz 10976  df-q 11068  df-rp 11106  df-xneg 11203  df-xadd 11204  df-xmul 11205  df-icc 11421  df-seq 11927  df-exp 11986  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-topgen 14504  df-psmet 17937  df-xmet 17938  df-met 17939  df-bl 17940  df-mopn 17941  df-top 18638  df-bases 18640  df-topon 18641  df-lm 18968  df-haus 19054  df-grpo 23850  df-gid 23851  df-ginv 23852  df-gdiv 23853  df-ablo 23941  df-subgo 23961  df-vc 24096  df-nv 24142  df-va 24145  df-ba 24146  df-sm 24147  df-0v 24148  df-vs 24149  df-nmcv 24150  df-ims 24151  df-ssp 24292  df-hnorm 24542  df-hba 24543  df-hvsub 24545  df-hlim 24546  df-sh 24781  df-ch 24796  df-ch0 24828
This theorem is referenced by:  hhsssh2  24843
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