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Theorem iscph 22196
Description: A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complex numbers, with a norm defined. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
iscph.v  |-  V  =  ( Base `  W
)
iscph.h  |-  .,  =  ( .i `  W )
iscph.n  |-  N  =  ( norm `  W
)
iscph.f  |-  F  =  (Scalar `  W )
iscph.k  |-  K  =  ( Base `  F
)
Assertion
Ref Expression
iscph  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) )
Distinct variable group:    x, W
Allowed substitution hints:    F( x)    ., ( x)    K( x)    N( x)    V( x)

Proof of Theorem iscph
Dummy variables  f 
k  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3628 . . . . 5  |-  ( W  e.  ( PreHil  i^i NrmMod )  <->  ( W  e.  PreHil  /\  W  e. NrmMod ) )
21anbi1i 706 . . . 4  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
3 df-3an 993 . . . 4  |-  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
42, 3bitr4i 260 . . 3  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
54anbi1i 706 . 2  |-  ( ( ( W  e.  (
PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  /\  (
( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) )
6 fvex 5897 . . . . . 6  |-  (Scalar `  w )  e.  _V
76a1i 11 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
8 fvex 5897 . . . . . . 7  |-  ( Base `  f )  e.  _V
98a1i 11 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
10 simplr 767 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  (Scalar `  w ) )
11 simpll 765 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  ->  w  =  W )
1211fveq2d 5891 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  (Scalar `  W )
)
13 iscph.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1412, 13syl6eqr 2513 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  F )
1510, 14eqtrd 2495 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
16 simpr 467 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  ( Base `  f ) )
1715fveq2d 5891 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  ( Base `  F
) )
18 iscph.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
1917, 18syl6eqr 2513 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  K )
2016, 19eqtrd 2495 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
2120oveq2d 6330 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
2215, 21eqeq12d 2476 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
2320ineq1d 3644 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  i^i  (
0 [,) +oo )
)  =  ( K  i^i  ( 0 [,) +oo ) ) )
2423imaeq2d 5186 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  =  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) ) )
2524, 20sseq12d 3472 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( ( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  C_  k  <->  ( sqr " ( K  i^i  (
0 [,) +oo )
) )  C_  K
) )
2611fveq2d 5891 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  ( norm `  W
) )
27 iscph.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2826, 27syl6eqr 2513 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  N )
2911fveq2d 5891 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  ( Base `  W
) )
30 iscph.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
3129, 30syl6eqr 2513 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  V )
3211fveq2d 5891 . . . . . . . . . . . . 13  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  ( .i
`  W ) )
33 iscph.h . . . . . . . . . . . . 13  |-  .,  =  ( .i `  W )
3432, 33syl6eqr 2513 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  .,  )
3534oveqd 6331 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( x ( .i
`  w ) x )  =  ( x 
.,  x ) )
3635fveq2d 5891 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr `  (
x ( .i `  w ) x ) )  =  ( sqr `  ( x  .,  x
) ) )
3731, 36mpteq12dv 4494 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( x  e.  (
Base `  w )  |->  ( sqr `  (
x ( .i `  w ) x ) ) )  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )
3828, 37eqeq12d 2476 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( ( norm `  w
)  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) )  <->  N  =  ( x  e.  V  |->  ( sqr `  (
x  .,  x )
) ) ) )
3922, 25, 383anbi123d 1348 . . . . . . 7  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( ( f  =  (flds  k )  /\  ( sqr " ( k  i^i  ( 0 [,) +oo ) ) )  C_  k  /\  ( norm `  w
)  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) )  <->  ( F  =  (flds  K )  /\  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) )
40 3anass 995 . . . . . . 7  |-  ( ( F  =  (flds  K )  /\  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )  <->  ( F  =  (flds  K )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) )
4139, 40syl6bb 269 . . . . . 6  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( ( f  =  (flds  k )  /\  ( sqr " ( k  i^i  ( 0 [,) +oo ) ) )  C_  k  /\  ( norm `  w
)  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) )  <->  ( F  =  (flds  K )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) ) )
429, 41sbcied 3315 . . . . 5  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( [. ( Base `  f
)  /  k ]. ( f  =  (flds  k )  /\  ( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  C_  k  /\  ( norm `  w )  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) )  <->  ( F  =  (flds  K )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) ) )
437, 42sbcied 3315 . . . 4  |-  ( w  =  W  ->  ( [. (Scalar `  w )  /  f ]. [. ( Base `  f )  / 
k ]. ( f  =  (flds  k )  /\  ( sqr " ( k  i^i  ( 0 [,) +oo ) ) )  C_  k  /\  ( norm `  w
)  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) )  <->  ( F  =  (flds  K )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) ) )
44 df-cph 22194 . . . 4  |-  CPreHil  =  {
w  e.  ( PreHil  i^i NrmMod )  |  [. (Scalar `  w )  /  f ]. [. ( Base `  f
)  /  k ]. ( f  =  (flds  k )  /\  ( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  C_  k  /\  ( norm `  w )  =  ( x  e.  ( Base `  w
)  |->  ( sqr `  (
x ( .i `  w ) x ) ) ) ) }
4543, 44elrab2 3209 . . 3  |-  ( W  e.  CPreHil 
<->  ( W  e.  (
PreHil  i^i NrmMod )  /\  ( F  =  (flds  K )  /\  (
( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) ) )
46 anass 659 . . 3  |-  ( ( ( W  e.  (
PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  /\  (
( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) )  <->  ( W  e.  ( PreHil  i^i NrmMod )  /\  ( F  =  (flds  K )  /\  (
( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) ) )
4745, 46bitr4i 260 . 2  |-  ( W  e.  CPreHil 
<->  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  /\  (
( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) )
48 3anass 995 . 2  |-  ( ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) ) )
495, 47, 483bitr4i 285 1  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  /\  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) )  C_  K  /\  N  =  ( x  e.  V  |->  ( sqr `  ( x 
.,  x ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897   _Vcvv 3056   [.wsbc 3278    i^i cin 3414    C_ wss 3415    |-> cmpt 4474   "cima 4855   ` cfv 5600  (class class class)co 6314   0cc0 9564   +oocpnf 9697   [,)cico 11665   sqrcsqrt 13344   Basecbs 15169   ↾s cress 15170  Scalarcsca 15241   .icip 15243  ℂfldccnfld 19018   PreHilcphl 19239   normcnm 21639  NrmModcnlm 21643   CPreHilccph 22192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-nul 4547
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-xp 4858  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fv 5608  df-ov 6317  df-cph 22194
This theorem is referenced by:  cphphl  22197  cphnlm  22198  cphsca  22205  cphsqrtcl  22210  cphnmfval  22218  tchcph  22259
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