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Theorem smfval 29836
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 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
smfval 𝑆 = (2nd ‘(1st𝑈))

Proof of Theorem smfval
StepHypRef Expression
1 smfval.4 . 2 𝑆 = ( ·𝑠OLD𝑈)
2 df-sm 29828 . . . . 5 ·𝑠OLD = (2nd ∘ 1st )
32fveq1i 6889 . . . 4 ( ·𝑠OLD𝑈) = ((2nd ∘ 1st )‘𝑈)
4 fo1st 7990 . . . . . 6 1st :V–onto→V
5 fof 6802 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6986 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
86, 7mpan 689 . . . 4 (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
93, 8eqtrid 2785 . . 3 (𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
10 fvprc 6880 . . . 4 𝑈 ∈ V → ( ·𝑠OLD𝑈) = ∅)
11 fvprc 6880 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6892 . . . . 5 𝑈 ∈ V → (2nd ‘(1st𝑈)) = (2nd ‘∅))
13 2nd0 7977 . . . . 5 (2nd ‘∅) = ∅
1412, 13eqtr2di 2790 . . . 4 𝑈 ∈ V → ∅ = (2nd ‘(1st𝑈)))
1510, 14eqtrd 2773 . . 3 𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
169, 15pm2.61i 182 . 2 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
171, 16eqtri 2761 1 𝑆 = (2nd ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4321  ccom 5679  wf 6536  ontowfo 6538  cfv 6540  1st c1st 7968  2nd c2nd 7969   ·𝑠OLD cns 29818
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 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7970  df-2nd 7971  df-sm 29828
This theorem is referenced by:  nvvop  29840  nvsf  29850  nvscl  29857  nvsid  29858  nvsass  29859  nvdi  29861  nvdir  29862  nv2  29863  nv0  29868  nvsz  29869  nvinv  29870  nvtri  29901  cnnvs  29911  phop  30049  ipdirilem  30060  h2hsm  30206  hhsssm  30489
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