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Theorem vafval 30632
Description: Value of the function for the vector addition (group) operation on a normed complex vector space. (Contributed by NM, 23-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
vafval.2 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
vafval 𝐺 = (1st ‘(1st𝑈))

Proof of Theorem vafval
StepHypRef Expression
1 vafval.2 . 2 𝐺 = ( +𝑣𝑈)
2 df-va 30624 . . . . 5 +𝑣 = (1st ∘ 1st )
32fveq1i 6908 . . . 4 ( +𝑣𝑈) = ((1st ∘ 1st )‘𝑈)
4 fo1st 8033 . . . . . 6 1st :V–onto→V
5 fof 6821 . . . . . 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) → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
86, 7mpan 690 . . . 4 (𝑈 ∈ V → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
93, 8eqtrid 2787 . . 3 (𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
10 fvprc 6899 . . . 4 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
11 fvprc 6899 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6911 . . . . 5 𝑈 ∈ V → (1st ‘(1st𝑈)) = (1st ‘∅))
13 1st0 8019 . . . . 5 (1st ‘∅) = ∅
1412, 13eqtr2di 2792 . . . 4 𝑈 ∈ V → ∅ = (1st ‘(1st𝑈)))
1510, 14eqtrd 2775 . . 3 𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
169, 15pm2.61i 182 . 2 ( +𝑣𝑈) = (1st ‘(1st𝑈))
171, 16eqtri 2763 1 𝐺 = (1st ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  ccom 5693  wf 6559  ontowfo 6561  cfv 6563  1st c1st 8011   +𝑣 cpv 30614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-va 30624
This theorem is referenced by:  nvvop  30638  nvablo  30645  nvsf  30648  nvscl  30655  nvsid  30656  nvsass  30657  nvdi  30659  nvdir  30660  nv2  30661  nv0  30666  nvsz  30667  nvinv  30668  cnnvg  30707  phop  30847  ip0i  30854  ipdirilem  30858  h2hva  31003  hhssva  31286  hhshsslem1  31296
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