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Theorem iscph 20669
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 3534 . . . . 5  |-  ( W  e.  ( PreHil  i^i NrmMod )  <->  ( W  e.  PreHil  /\  W  e. NrmMod ) )
21anbi1i 695 . . . 4  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
3 df-3an 967 . . . 4  |-  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
42, 3bitr4i 252 . . 3  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
54anbi1i 695 . 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 5696 . . . . . 6  |-  (Scalar `  w )  e.  _V
76a1i 11 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
8 fvex 5696 . . . . . . 7  |-  ( Base `  f )  e.  _V
98a1i 11 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
10 simplr 754 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  (Scalar `  w ) )
11 simpll 753 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  ->  w  =  W )
1211fveq2d 5690 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  (Scalar `  W )
)
13 iscph.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1412, 13syl6eqr 2488 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  F )
1510, 14eqtrd 2470 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
16 simpr 461 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  ( Base `  f ) )
1715fveq2d 5690 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  ( Base `  F
) )
18 iscph.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
1917, 18syl6eqr 2488 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  K )
2016, 19eqtrd 2470 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
2120oveq2d 6102 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
2215, 21eqeq12d 2452 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
2320ineq1d 3546 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  i^i  (
0 [,) +oo )
)  =  ( K  i^i  ( 0 [,) +oo ) ) )
2423imaeq2d 5164 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  =  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) ) )
2524, 20sseq12d 3380 . . . . . . . 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 5690 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  ( norm `  W
) )
27 iscph.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2826, 27syl6eqr 2488 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  N )
2911fveq2d 5690 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  ( Base `  W
) )
30 iscph.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
3129, 30syl6eqr 2488 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  V )
3211fveq2d 5690 . . . . . . . . . . . . 13  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  ( .i
`  W ) )
33 iscph.h . . . . . . . . . . . . 13  |-  .,  =  ( .i `  W )
3432, 33syl6eqr 2488 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  .,  )
3534oveqd 6103 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( x ( .i
`  w ) x )  =  ( x 
.,  x ) )
3635fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr `  (
x ( .i `  w ) x ) )  =  ( sqr `  ( x  .,  x
) ) )
3731, 36mpteq12dv 4365 . . . . . . . . 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 2452 . . . . . . . 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 1289 . . . . . . 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 969 . . . . . . 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 261 . . . . . 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 3218 . . . . 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 3218 . . . 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 20667 . . . 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 3114 . . 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 649 . . 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 252 . 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 969 . 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 277 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 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2967   [.wsbc 3181    i^i cin 3322    C_ wss 3323    e. cmpt 4345   "cima 4838   ` cfv 5413  (class class class)co 6086   0cc0 9274   +oocpnf 9407   [,)cico 11294   sqrcsqr 12714   Basecbs 14166   ↾s cress 14167  Scalarcsca 14233   .icip 14235  ℂfldccnfld 17798   PreHilcphl 18033   normcnm 20149  NrmModcnlm 20153   CPreHilccph 20665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-nul 4416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-xp 4841  df-cnv 4843  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fv 5421  df-ov 6089  df-cph 20667
This theorem is referenced by:  cphphl  20670  cphnlm  20671  cphsca  20678  cphsqrcl  20683  cphnmfval  20691  tchcph  20732
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