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Theorem iscph 22090
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 3592 . . . . 5  |-  ( W  e.  ( PreHil  i^i NrmMod )  <->  ( W  e.  PreHil  /\  W  e. NrmMod ) )
21anbi1i 699 . . . 4  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
3 df-3an 984 . . . 4  |-  ( ( W  e.  PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) )  <->  ( ( W  e.  PreHil  /\  W  e. NrmMod )  /\  F  =  (flds  K ) ) )
42, 3bitr4i 255 . . 3  |-  ( ( W  e.  ( PreHil  i^i NrmMod )  /\  F  =  (flds  K ) )  <->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  F  =  (flds  K ) ) )
54anbi1i 699 . 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 5835 . . . . . 6  |-  (Scalar `  w )  e.  _V
76a1i 11 . . . . 5  |-  ( w  =  W  ->  (Scalar `  w )  e.  _V )
8 fvex 5835 . . . . . . 7  |-  ( Base `  f )  e.  _V
98a1i 11 . . . . . 6  |-  ( ( w  =  W  /\  f  =  (Scalar `  w
) )  ->  ( Base `  f )  e. 
_V )
10 simplr 760 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  (Scalar `  w ) )
11 simpll 758 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  ->  w  =  W )
1211fveq2d 5829 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  (Scalar `  W )
)
13 iscph.f . . . . . . . . . . 11  |-  F  =  (Scalar `  W )
1412, 13syl6eqr 2480 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(Scalar `  w )  =  F )
1510, 14eqtrd 2462 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
f  =  F )
16 simpr 462 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  ( Base `  f ) )
1715fveq2d 5829 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  ( Base `  F
) )
18 iscph.k . . . . . . . . . . . 12  |-  K  =  ( Base `  F
)
1917, 18syl6eqr 2480 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  f )  =  K )
2016, 19eqtrd 2462 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
k  =  K )
2120oveq2d 6265 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
(flds  k
)  =  (flds  K ) )
2215, 21eqeq12d 2443 . . . . . . . 8  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( f  =  (flds  k )  <-> 
F  =  (flds  K ) ) )
2320ineq1d 3606 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( k  i^i  (
0 [,) +oo )
)  =  ( K  i^i  ( 0 [,) +oo ) ) )
2423imaeq2d 5130 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr " (
k  i^i  ( 0 [,) +oo ) ) )  =  ( sqr " ( K  i^i  ( 0 [,) +oo ) ) ) )
2524, 20sseq12d 3436 . . . . . . . 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 5829 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  ( norm `  W
) )
27 iscph.n . . . . . . . . . 10  |-  N  =  ( norm `  W
)
2826, 27syl6eqr 2480 . . . . . . . . 9  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( norm `  w )  =  N )
2911fveq2d 5829 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  ( Base `  W
) )
30 iscph.v . . . . . . . . . . 11  |-  V  =  ( Base `  W
)
3129, 30syl6eqr 2480 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( Base `  w )  =  V )
3211fveq2d 5829 . . . . . . . . . . . . 13  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  ( .i
`  W ) )
33 iscph.h . . . . . . . . . . . . 13  |-  .,  =  ( .i `  W )
3432, 33syl6eqr 2480 . . . . . . . . . . . 12  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( .i `  w
)  =  .,  )
3534oveqd 6266 . . . . . . . . . . 11  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( x ( .i
`  w ) x )  =  ( x 
.,  x ) )
3635fveq2d 5829 . . . . . . . . . 10  |-  ( ( ( w  =  W  /\  f  =  (Scalar `  w ) )  /\  k  =  ( Base `  f ) )  -> 
( sqr `  (
x ( .i `  w ) x ) )  =  ( sqr `  ( x  .,  x
) ) )
3731, 36mpteq12dv 4445 . . . . . . . . 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 2443 . . . . . . . 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 1335 . . . . . . 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 986 . . . . . . 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 264 . . . . . 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 3279 . . . . 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 3279 . . . 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 22088 . . . 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 3173 . . 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 653 . . 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 255 . 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 986 . 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 280 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   _Vcvv 3022   [.wsbc 3242    i^i cin 3378    C_ wss 3379    |-> cmpt 4425   "cima 4799   ` cfv 5544  (class class class)co 6249   0cc0 9490   +oocpnf 9623   [,)cico 11588   sqrcsqrt 13240   Basecbs 15064   ↾s cress 15065  Scalarcsca 15136   .icip 15138  ℂfldccnfld 18913   PreHilcphl 19133   normcnm 21533  NrmModcnlm 21537   CPreHilccph 22086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-nul 4498
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-xp 4802  df-cnv 4804  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fv 5552  df-ov 6252  df-cph 22088
This theorem is referenced by:  cphphl  22091  cphnlm  22092  cphsca  22099  cphsqrtcl  22104  cphnmfval  22112  tchcph  22153
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