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Theorem cphnlm 19136
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 2437 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2437 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2437 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2437 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2437 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 19134 . . 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 973 . 2  |-  ( W  e.  CPreHil  ->  ( W  e. 
PreHil  /\  W  e. NrmMod  /\  (Scalar `  W )  =  (flds  ( Base `  (Scalar `  W )
) ) ) )
87simp2d 971 1  |-  ( W  e.  CPreHil  ->  W  e. NrmMod )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3320    C_ wss 3321    e. cmpt 4267   "cima 4882   ` cfv 5455  (class class class)co 6082   0cc0 8991    +oocpnf 9118   [,)cico 10919   sqrcsqr 12039   Basecbs 13470   ↾s cress 13471  Scalarcsca 13533   .icip 13535  ℂfldccnfld 16704   PreHilcphl 16856   normcnm 18625  NrmModcnlm 18629   CPreHilccph 19130
This theorem is referenced by:  cphngp  19137  cphlmod  19138  cphnvc  19140  cphnmvs  19154  ipcnlem2  19199  ipcnlem1  19200  csscld  19204
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-nul 4339
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-xp 4885  df-cnv 4887  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fv 5463  df-ov 6085  df-cph 19132
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