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Theorem cphnlm 22228
Description: A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphnlm  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )

Proof of Theorem cphnlm
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2471 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2471 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2471 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2471 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2471 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 22226 . . 3  |-  ( W  e.  CPreHil 
<->  ( ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) )  /\  ( sqr " ( (
Base `  (Scalar `  W
) )  i^i  (
0 [,) +oo )
) )  C_  ( Base `  (Scalar `  W
) )  /\  ( norm `  W )  =  ( x  e.  (
Base `  W )  |->  ( sqr `  (
x ( .i `  W ) x ) ) ) ) )
76simp1bi 1045 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp2d 1043 1  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904    i^i cin 3389    C_ wss 3390    |-> cmpt 4454   "cima 4842   ` cfv 5589  (class class class)co 6308   0cc0 9557   +oocpnf 9690   [,)cico 11662   sqrcsqrt 13373   Basecbs 15199   ↾s cress 15200  Scalarcsca 15271   .icip 15273  ℂfldccnfld 19047   PreHilcphl 19268   normcnm 21669  NrmModcnlm 21673   CPreHilccph 22222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-nul 4527
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fv 5597  df-ov 6311  df-cph 22224
This theorem is referenced by:  cphngp  22229  cphlmod  22230  cphnvc  22232  cphnmvs  22246  ipcnlem2  22293  ipcnlem1  22294  csscld  22298
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