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Theorem cphnlm 21349
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 2462 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2462 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2462 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2462 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2462 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 21347 . . 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 1006 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp2d 1004 1  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762    i^i cin 3470    C_ wss 3471    |-> cmpt 4500   "cima 4997   ` cfv 5581  (class class class)co 6277   0cc0 9483   +oocpnf 9616   [,)cico 11522   sqrcsqr 13018   Basecbs 14481   ↾s cress 14482  Scalarcsca 14549   .icip 14551  ℂfldccnfld 18186   PreHilcphl 18421   normcnm 20827  NrmModcnlm 20831   CPreHilccph 21343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-nul 4571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fv 5589  df-ov 6280  df-cph 21345
This theorem is referenced by:  cphngp  21350  cphlmod  21351  cphnvc  21353  cphnmvs  21367  ipcnlem2  21414  ipcnlem1  21415  csscld  21419
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