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Theorem vafval 30485
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 30477 . . . . 5 +𝑣 = (1st ∘ 1st )
32fveq1i 6897 . . . 4 ( +𝑣𝑈) = ((1st ∘ 1st )‘𝑈)
4 fo1st 8014 . . . . . 6 1st :V–onto→V
5 fof 6810 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6996 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
86, 7mpan 688 . . . 4 (𝑈 ∈ V → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
93, 8eqtrid 2777 . . 3 (𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
10 fvprc 6888 . . . 4 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
11 fvprc 6888 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6900 . . . . 5 𝑈 ∈ V → (1st ‘(1st𝑈)) = (1st ‘∅))
13 1st0 8000 . . . . 5 (1st ‘∅) = ∅
1412, 13eqtr2di 2782 . . . 4 𝑈 ∈ V → ∅ = (1st ‘(1st𝑈)))
1510, 14eqtrd 2765 . . 3 𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
169, 15pm2.61i 182 . 2 ( +𝑣𝑈) = (1st ‘(1st𝑈))
171, 16eqtri 2753 1 𝐺 = (1st ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3461  c0 4322  ccom 5682  wf 6545  ontowfo 6547  cfv 6549  1st c1st 7992   +𝑣 cpv 30467
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fo 6555  df-fv 6557  df-1st 7994  df-va 30477
This theorem is referenced by:  nvvop  30491  nvablo  30498  nvsf  30501  nvscl  30508  nvsid  30509  nvsass  30510  nvdi  30512  nvdir  30513  nv2  30514  nv0  30519  nvsz  30520  nvinv  30521  cnnvg  30560  phop  30700  ip0i  30707  ipdirilem  30711  h2hva  30856  hhssva  31139  hhshsslem1  31149
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