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Theorem iscph 19086
Description: A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complexes, 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 3490 . . . . 5  |-  ( W  e.  ( PreHil  i^i NrmMod )  <->  ( W  e.  PreHil  /\  W  e. NrmMod ) )
21anbi1i 677 . . . 4  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
3 df-3an 938 . . . 4  |-  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
42, 3bitr4i 244 . . 3  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
54anbi1i 677 . 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 5701 . . . . . 6  |-  (Scalar `  w )  e.  _V
76a1i 11 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
8 fvex 5701 . . . . . . 7  |-  ( Base `  f )  e.  _V
98a1i 11 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
10 simplr 732 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  (Scalar `  w ) )
11 simpll 731 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  ->  w  =  W )
1211fveq2d 5691 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  (Scalar `  W )
)
13 iscph.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1412, 13syl6eqr 2454 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  F )
1510, 14eqtrd 2436 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
16 simpr 448 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  ( Base `  f ) )
1715fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  ( Base `  F
) )
18 iscph.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
1917, 18syl6eqr 2454 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  K )
2016, 19eqtrd 2436 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
2120oveq2d 6056 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
2215, 21eqeq12d 2418 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
2320ineq1d 3501 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  i^i  (
0 [,)  +oo ) )  =  ( K  i^i  ( 0 [,)  +oo ) ) )
2423imaeq2d 5162 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr " (
k  i^i  ( 0 [,)  +oo ) ) )  =  ( sqr " ( K  i^i  ( 0 [,) 
+oo ) ) ) )
2524, 20sseq12d 3337 . . . . . . . 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 5691 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  ( norm `  W
) )
27 iscph.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2826, 27syl6eqr 2454 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  N )
2911fveq2d 5691 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  ( Base `  W
) )
30 iscph.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
3129, 30syl6eqr 2454 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  V )
3211fveq2d 5691 . . . . . . . . . . . . 13  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  ( .i
`  W ) )
33 iscph.h . . . . . . . . . . . . 13  |-  .,  =  ( .i `  W )
3432, 33syl6eqr 2454 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  .,  )
3534oveqd 6057 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( x ( .i
`  w ) x )  =  ( x 
.,  x ) )
3635fveq2d 5691 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr `  (
x ( .i `  w ) x ) )  =  ( sqr `  ( x  .,  x
) ) )
3731, 36mpteq12dv 4247 . . . . . . . . 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 2418 . . . . . . . 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 1254 . . . . . . 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 940 . . . . . . 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 253 . . . . . 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 3157 . . . . 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 3157 . . . 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 19084 . . . 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 3054 . . 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 631 . . 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 244 . 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 940 . 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 269 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 set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   [.wsbc 3121    i^i cin 3279    C_ wss 3280    e. cmpt 4226   "cima 4840   ` cfv 5413  (class class class)co 6040   0cc0 8946    +oocpnf 9073   [,)cico 10874   sqrcsqr 11993   Basecbs 13424   ↾s cress 13425  Scalarcsca 13487   .icip 13489  ℂfldccnfld 16658   PreHilcphl 16810   normcnm 18577  NrmModcnlm 18581   CPreHilccph 19082
This theorem is referenced by:  cphphl  19087  cphnlm  19088  cphsca  19095  cphsqrcl  19100  cphnmfval  19108  tchcph  19147
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-ov 6043  df-cph 19084
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