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Theorem smfval 28501
 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 28493 . . . . 5 ·𝑠OLD = (2nd ∘ 1st )
32fveq1i 6664 . . . 4 ( ·𝑠OLD𝑈) = ((2nd ∘ 1st )‘𝑈)
4 fo1st 7719 . . . . . 6 1st :V–onto→V
5 fof 6581 . . . . . 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 689 . . . 4 (𝑈 ∈ V → ((2nd ∘ 1st )‘𝑈) = (2nd ‘(1st𝑈)))
93, 8syl5eq 2805 . . 3 (𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
10 fvprc 6655 . . . 4 𝑈 ∈ V → ( ·𝑠OLD𝑈) = ∅)
11 fvprc 6655 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6667 . . . . 5 𝑈 ∈ V → (2nd ‘(1st𝑈)) = (2nd ‘∅))
13 2nd0 7706 . . . . 5 (2nd ‘∅) = ∅
1412, 13eqtr2di 2810 . . . 4 𝑈 ∈ V → ∅ = (2nd ‘(1st𝑈)))
1510, 14eqtrd 2793 . . 3 𝑈 ∈ V → ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈)))
169, 15pm2.61i 185 . 2 ( ·𝑠OLD𝑈) = (2nd ‘(1st𝑈))
171, 16eqtri 2781 1 𝑆 = (2nd ‘(1st𝑈))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227   ∘ ccom 5532  ⟶wf 6336  –onto→wfo 6338  ‘cfv 6340  1st c1st 7697  2nd c2nd 7698   ·𝑠OLD cns 28483 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-1st 7699  df-2nd 7700  df-sm 28493 This theorem is referenced by:  nvvop  28505  nvsf  28515  nvscl  28522  nvsid  28523  nvsass  28524  nvdi  28526  nvdir  28527  nv2  28528  nv0  28533  nvsz  28534  nvinv  28535  nvtri  28566  cnnvs  28576  phop  28714  ipdirilem  28725  h2hsm  28871  hhsssm  29154
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