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Theorem nvsid 25195
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 2467 . . 3  |-  ( 1st `  U )  =  ( 1st `  U )
21nvvc 25181 . 2  |-  ( U  e.  NrmCVec  ->  ( 1st `  U
)  e.  CVecOLD )
3 eqid 2467 . . . 4  |-  ( +v
`  U )  =  ( +v `  U
)
43vafval 25169 . . 3  |-  ( +v
`  U )  =  ( 1st `  ( 1st `  U ) )
5 nvscl.4 . . . 4  |-  S  =  ( .sOLD `  U )
65smfval 25171 . . 3  |-  S  =  ( 2nd `  ( 1st `  U ) )
7 nvscl.1 . . . 4  |-  X  =  ( BaseSet `  U )
87, 3bafval 25170 . . 3  |-  X  =  ran  ( +v `  U )
94, 6, 8vcid 25117 . 2  |-  ( ( ( 1st `  U
)  e.  CVecOLD  /\  A  e.  X )  ->  ( 1 S A )  =  A )
102, 9sylan 471 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   1stc1st 6779   1c1 9489   CVecOLDcvc 25111   NrmCVeccnv 25150   +vcpv 25151   BaseSetcba 25152   .sOLDcns 25153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-1st 6781  df-2nd 6782  df-vc 25112  df-nv 25158  df-va 25161  df-ba 25162  df-sm 25163  df-0v 25164  df-nmcv 25166
This theorem is referenced by:  nvmul0or  25220  nvnncan  25231  nvpi  25242  nvge0  25250  ipval2lem3  25288  ipval2  25290  ipval2lem6  25294  ipidsq  25296  lnoadd  25346  ip1ilem  25414  ip2i  25416  ipdirilem  25417  ipasslem1  25419  ipasslem4  25422  ipasslem10  25427  minvecolem2  25464  hlmulid  25494
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