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Theorem smfval 27765
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 27757 . . . . 5 ·𝑠OLD = (2nd ∘ 1st )
32fveq1i 6349 . . . 4 ( ·𝑠OLD𝑈) = ((2nd ∘ 1st )‘𝑈)
4 fo1st 7349 . . . . . 6 1st :V–onto→V
5 fof 6272 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6433 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
86, 7mpan 708 . . . 4 (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
93, 8syl5eq 2802 . . 3 (𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
10 fvprc 6342 . . . 4 𝑈 ∈ V → ( ·𝑠OLD𝑈) = ∅)
11 fvprc 6342 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6352 . . . . 5 𝑈 ∈ V → (2nd ‘(1st𝑈)) = (2nd ‘∅))
13 2nd0 7336 . . . . 5 (2nd ‘∅) = ∅
1412, 13syl6req 2807 . . . 4 𝑈 ∈ V → ∅ = (2nd ‘(1st𝑈)))
1510, 14eqtrd 2790 . . 3 𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
169, 15pm2.61i 176 . 2 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
171, 16eqtri 2778 1 𝑆 = (2nd ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1628  wcel 2135  Vcvv 3336  c0 4054  ccom 5266  wf 6041  ontowfo 6043  cfv 6045  1st c1st 7327  2nd c2nd 7328   ·𝑠OLD cns 27747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-fo 6051  df-fv 6053  df-1st 7329  df-2nd 7330  df-sm 27757
This theorem is referenced by:  nvvop  27769  nvsf  27779  nvscl  27786  nvsid  27787  nvsass  27788  nvdi  27790  nvdir  27791  nv2  27792  nv0  27797  nvsz  27798  nvinv  27799  nvtri  27830  cnnvs  27840  phop  27978  phpar  27984  ipdirilem  27989  h2hsm  28137  hhsssm  28420
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