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Theorem nvsid 25936
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  =  ( .sOLD `  U )
Assertion
Ref Expression
nvsid  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )

Proof of Theorem nvsid
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25922 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2402 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25910 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25912 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25911 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vcid 25858 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
102, 9sylan 469 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   ` cfv 5569  (class class class)co 6278   1stc1st 6782   1c1 9523   CVecOLDcvc 25852   NrmCVeccnv 25891   +vcpv 25892   BaseSetcba 25893   .sOLDcns 25894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-1st 6784  df-2nd 6785  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-nmcv 25907
This theorem is referenced by:  nvmul0or  25961  nvnncan  25972  nvpi  25983  nvge0  25991  ipval2lem3  26029  ipval2  26031  ipval2lem6  26035  ipidsq  26037  lnoadd  26087  ip1ilem  26155  ip2i  26157  ipdirilem  26158  ipasslem1  26160  ipasslem4  26163  ipasslem10  26168  minvecolem2  26205  hlmulid  26235
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