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Theorem smfval 29858
Description: Value of the function for the scalar multiplication operation on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
smfval.4 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
Assertion
Ref Expression
smfval ๐‘† = (2nd โ€˜(1st โ€˜๐‘ˆ))

Proof of Theorem smfval
StepHypRef Expression
1 smfval.4 . 2 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
2 df-sm 29850 . . . . 5 ยท๐‘ OLD = (2nd โˆ˜ 1st )
32fveq1i 6893 . . . 4 ( ยท๐‘ OLD โ€˜๐‘ˆ) = ((2nd โˆ˜ 1st )โ€˜๐‘ˆ)
4 fo1st 7995 . . . . . 6 1st :Vโ€“ontoโ†’V
5 fof 6806 . . . . . 6 (1st :Vโ€“ontoโ†’V โ†’ 1st :VโŸถV)
64, 5ax-mp 5 . . . . 5 1st :VโŸถV
7 fvco3 6991 . . . . 5 ((1st :VโŸถV โˆง ๐‘ˆ โˆˆ V) โ†’ ((2nd โˆ˜ 1st )โ€˜๐‘ˆ) = (2nd โ€˜(1st โ€˜๐‘ˆ)))
86, 7mpan 689 . . . 4 (๐‘ˆ โˆˆ V โ†’ ((2nd โˆ˜ 1st )โ€˜๐‘ˆ) = (2nd โ€˜(1st โ€˜๐‘ˆ)))
93, 8eqtrid 2785 . . 3 (๐‘ˆ โˆˆ V โ†’ ( ยท๐‘ OLD โ€˜๐‘ˆ) = (2nd โ€˜(1st โ€˜๐‘ˆ)))
10 fvprc 6884 . . . 4 (ยฌ ๐‘ˆ โˆˆ V โ†’ ( ยท๐‘ OLD โ€˜๐‘ˆ) = โˆ…)
11 fvprc 6884 . . . . . 6 (ยฌ ๐‘ˆ โˆˆ V โ†’ (1st โ€˜๐‘ˆ) = โˆ…)
1211fveq2d 6896 . . . . 5 (ยฌ ๐‘ˆ โˆˆ V โ†’ (2nd โ€˜(1st โ€˜๐‘ˆ)) = (2nd โ€˜โˆ…))
13 2nd0 7982 . . . . 5 (2nd โ€˜โˆ…) = โˆ…
1412, 13eqtr2di 2790 . . . 4 (ยฌ ๐‘ˆ โˆˆ V โ†’ โˆ… = (2nd โ€˜(1st โ€˜๐‘ˆ)))
1510, 14eqtrd 2773 . . 3 (ยฌ ๐‘ˆ โˆˆ V โ†’ ( ยท๐‘ OLD โ€˜๐‘ˆ) = (2nd โ€˜(1st โ€˜๐‘ˆ)))
169, 15pm2.61i 182 . 2 ( ยท๐‘ OLD โ€˜๐‘ˆ) = (2nd โ€˜(1st โ€˜๐‘ˆ))
171, 16eqtri 2761 1 ๐‘† = (2nd โ€˜(1st โ€˜๐‘ˆ))
Colors of variables: wff setvar class
Syntax hints:  ยฌ wn 3   = wceq 1542   โˆˆ wcel 2107  Vcvv 3475  โˆ…c0 4323   โˆ˜ ccom 5681  โŸถwf 6540  โ€“ontoโ†’wfo 6542  โ€˜cfv 6544  1st c1st 7973  2nd c2nd 7974   ยท๐‘ OLD cns 29840
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-2nd 7976  df-sm 29850
This theorem is referenced by:  nvvop  29862  nvsf  29872  nvscl  29879  nvsid  29880  nvsass  29881  nvdi  29883  nvdir  29884  nv2  29885  nv0  29890  nvsz  29891  nvinv  29892  nvtri  29923  cnnvs  29933  phop  30071  ipdirilem  30082  h2hsm  30228  hhsssm  30511
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