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Theorem smfval 28296
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 28288 . . . . 5 ·𝑠OLD = (2nd ∘ 1st )
32fveq1i 6667 . . . 4 ( ·𝑠OLD𝑈) = ((2nd ∘ 1st )‘𝑈)
4 fo1st 7703 . . . . . 6 1st :V–onto→V
5 fof 6586 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6756 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
86, 7mpan 686 . . . 4 (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
93, 8syl5eq 2872 . . 3 (𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
10 fvprc 6659 . . . 4 𝑈 ∈ V → ( ·𝑠OLD𝑈) = ∅)
11 fvprc 6659 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6670 . . . . 5 𝑈 ∈ V → (2nd ‘(1st𝑈)) = (2nd ‘∅))
13 2nd0 7690 . . . . 5 (2nd ‘∅) = ∅
1412, 13syl6req 2877 . . . 4 𝑈 ∈ V → ∅ = (2nd ‘(1st𝑈)))
1510, 14eqtrd 2860 . . 3 𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
169, 15pm2.61i 183 . 2 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
171, 16eqtri 2848 1 𝑆 = (2nd ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3499  c0 4294  ccom 5557  wf 6347  ontowfo 6349  cfv 6351  1st c1st 7681  2nd c2nd 7682   ·𝑠OLD cns 28278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-sm 28288
This theorem is referenced by:  nvvop  28300  nvsf  28310  nvscl  28317  nvsid  28318  nvsass  28319  nvdi  28321  nvdir  28322  nv2  28323  nv0  28328  nvsz  28329  nvinv  28330  nvtri  28361  cnnvs  28371  phop  28509  ipdirilem  28520  h2hsm  28666  hhsssm  28949
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