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Theorem vafval 29856
Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
vafval.2 šŗ = ( +š‘£ ā€˜š‘ˆ)
Assertion
Ref Expression
vafval šŗ = (1st ā€˜(1st ā€˜š‘ˆ))

Proof of Theorem vafval
StepHypRef Expression
1 vafval.2 . 2 šŗ = ( +š‘£ ā€˜š‘ˆ)
2 df-va 29848 . . . . 5 +š‘£ = (1st āˆ˜ 1st )
32fveq1i 6893 . . . 4 ( +š‘£ ā€˜š‘ˆ) = ((1st āˆ˜ 1st )ā€˜š‘ˆ)
4 fo1st 7995 . . . . . 6 1st :Vā€“ontoā†’V
5 fof 6806 . . . . . 6 (1st :Vā€“ontoā†’V ā†’ 1st :VāŸ¶V)
64, 5ax-mp 5 . . . . 5 1st :VāŸ¶V
7 fvco3 6991 . . . . 5 ((1st :VāŸ¶V āˆ§ š‘ˆ āˆˆ V) ā†’ ((1st āˆ˜ 1st )ā€˜š‘ˆ) = (1st ā€˜(1st ā€˜š‘ˆ)))
86, 7mpan 689 . . . 4 (š‘ˆ āˆˆ V ā†’ ((1st āˆ˜ 1st )ā€˜š‘ˆ) = (1st ā€˜(1st ā€˜š‘ˆ)))
93, 8eqtrid 2785 . . 3 (š‘ˆ āˆˆ V ā†’ ( +š‘£ ā€˜š‘ˆ) = (1st ā€˜(1st ā€˜š‘ˆ)))
10 fvprc 6884 . . . 4 (Ā¬ š‘ˆ āˆˆ V ā†’ ( +š‘£ ā€˜š‘ˆ) = āˆ…)
11 fvprc 6884 . . . . . 6 (Ā¬ š‘ˆ āˆˆ V ā†’ (1st ā€˜š‘ˆ) = āˆ…)
1211fveq2d 6896 . . . . 5 (Ā¬ š‘ˆ āˆˆ V ā†’ (1st ā€˜(1st ā€˜š‘ˆ)) = (1st ā€˜āˆ…))
13 1st0 7981 . . . . 5 (1st ā€˜āˆ…) = āˆ…
1412, 13eqtr2di 2790 . . . 4 (Ā¬ š‘ˆ āˆˆ V ā†’ āˆ… = (1st ā€˜(1st ā€˜š‘ˆ)))
1510, 14eqtrd 2773 . . 3 (Ā¬ š‘ˆ āˆˆ V ā†’ ( +š‘£ ā€˜š‘ˆ) = (1st ā€˜(1st ā€˜š‘ˆ)))
169, 15pm2.61i 182 . 2 ( +š‘£ ā€˜š‘ˆ) = (1st ā€˜(1st ā€˜š‘ˆ))
171, 16eqtri 2761 1 šŗ = (1st ā€˜(1st ā€˜š‘ˆ))
Colors of variables: wff setvar class
Syntax hints:  Ā¬ wn 3   = wceq 1542   āˆˆ wcel 2107  Vcvv 3475  āˆ…c0 4323   āˆ˜ ccom 5681  āŸ¶wf 6540  ā€“ontoā†’wfo 6542  ā€˜cfv 6544  1st c1st 7973   +š‘£ cpv 29838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-va 29848
This theorem is referenced by:  nvvop  29862  nvablo  29869  nvsf  29872  nvscl  29879  nvsid  29880  nvsass  29881  nvdi  29883  nvdir  29884  nv2  29885  nv0  29890  nvsz  29891  nvinv  29892  cnnvg  29931  phop  30071  ip0i  30078  ipdirilem  30082  h2hva  30227  hhssva  30510  hhshsslem1  30520
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