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Theorem smfval 30624
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 30616 . . . . 5 ·𝑠OLD = (2nd ∘ 1st )
32fveq1i 6907 . . . 4 ( ·𝑠OLD𝑈) = ((2nd ∘ 1st )‘𝑈)
4 fo1st 8034 . . . . . 6 1st :V–onto→V
5 fof 6820 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 7008 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
86, 7mpan 690 . . . 4 (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
93, 8eqtrid 2789 . . 3 (𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
10 fvprc 6898 . . . 4 𝑈 ∈ V → ( ·𝑠OLD𝑈) = ∅)
11 fvprc 6898 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6910 . . . . 5 𝑈 ∈ V → (2nd ‘(1st𝑈)) = (2nd ‘∅))
13 2nd0 8021 . . . . 5 (2nd ‘∅) = ∅
1412, 13eqtr2di 2794 . . . 4 𝑈 ∈ V → ∅ = (2nd ‘(1st𝑈)))
1510, 14eqtrd 2777 . . 3 𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
169, 15pm2.61i 182 . 2 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
171, 16eqtri 2765 1 𝑆 = (2nd ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  ccom 5689  wf 6557  ontowfo 6559  cfv 6561  1st c1st 8012  2nd c2nd 8013   ·𝑠OLD cns 30606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-sm 30616
This theorem is referenced by:  nvvop  30628  nvsf  30638  nvscl  30645  nvsid  30646  nvsass  30647  nvdi  30649  nvdir  30650  nv2  30651  nv0  30656  nvsz  30657  nvinv  30658  nvtri  30689  cnnvs  30699  phop  30837  ipdirilem  30848  h2hsm  30994  hhsssm  31277
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