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Theorem cphnlm 18608
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 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2283 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2283 . . . 4  |-  ( norm `  W )  =  (
norm `  W )
4 eqid 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
5 eqid 2283 . . . 4  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
61, 2, 3, 4, 5iscph 18606 . . 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 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152    e. cmpt 4077   "cima 4692   ` cfv 5255  (class class class)co 5858   0cc0 8737    +oocpnf 8864   [,)cico 10658   sqrcsqr 11718   Basecbs 13148   ↾s cress 13149  Scalarcsca 13211   .icip 13213  ℂfldccnfld 16377   PreHilcphl 16528   normcnm 18099  NrmModcnlm 18103   CPreHilccph 18602
This theorem is referenced by:  cphngp  18609  cphlmod  18610  cphnvc  18612  cphnmvs  18626  ipcnlem2  18671  ipcnlem1  18672  csscld  18676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-ov 5861  df-cph 18604
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