MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvsid Structured version   Visualization version   GIF version

Theorem nvsid 28007
Description: Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1 𝑋 = (BaseSet‘𝑈)
nvscl.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvsid ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Proof of Theorem nvsid
StepHypRef Expression
1 eqid 2799 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 27995 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2799 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 27983 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 27985 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 27984 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcidOLD 27944 . 2 (((1st𝑈) ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
102, 9sylan 576 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  1st c1st 7399  1c1 10225  CVecOLDcvc 27938  NrmCVeccnv 27964   +𝑣 cpv 27965  BaseSetcba 27966   ·𝑠OLD cns 27967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-1st 7401  df-2nd 7402  df-vc 27939  df-nv 27972  df-va 27975  df-ba 27976  df-sm 27977  df-0v 27978  df-nmcv 27980
This theorem is referenced by:  nvmul0or  28030  nvpi  28047  nvge0  28053  ipval2lem3  28085  ipval2  28087  ipidsq  28090  lnoadd  28138  ip1ilem  28206  ip2i  28208  ipdirilem  28209  ipasslem1  28211  ipasslem4  28214  ipasslem10  28219  minvecolem2  28256  hlmulid  28286
  Copyright terms: Public domain W3C validator