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Theorem cphnlm 18624
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 2296 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2296 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2296 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2296 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 18622 . . 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 970 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp2d 968 1  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165    e. cmpt 4093   "cima 4708   ` cfv 5271  (class class class)co 5874   0cc0 8753    +oocpnf 8880   [,)cico 10674   sqrcsqr 11734   Basecbs 13164   ↾s cress 13165  Scalarcsca 13227   .icip 13229  ℂfldccnfld 16393   PreHilcphl 16544   normcnm 18115  NrmModcnlm 18119   CPreHilccph 18618
This theorem is referenced by:  cphngp  18625  cphlmod  18626  cphnvc  18628  cphnmvs  18642  ipcnlem2  18687  ipcnlem1  18688  csscld  18692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-cph 18620
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