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Theorem vafval 28400
 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 28392 . . . . 5 +𝑣 = (1st ∘ 1st )
32fveq1i 6651 . . . 4 ( +𝑣𝑈) = ((1st ∘ 1st )‘𝑈)
4 fo1st 7698 . . . . . 6 1st :V–onto→V
5 fof 6568 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6742 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
86, 7mpan 689 . . . 4 (𝑈 ∈ V → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
93, 8syl5eq 2845 . . 3 (𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
10 fvprc 6642 . . . 4 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
11 fvprc 6642 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6654 . . . . 5 𝑈 ∈ V → (1st ‘(1st𝑈)) = (1st ‘∅))
13 1st0 7684 . . . . 5 (1st ‘∅) = ∅
1412, 13eqtr2di 2850 . . . 4 𝑈 ∈ V → ∅ = (1st ‘(1st𝑈)))
1510, 14eqtrd 2833 . . 3 𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
169, 15pm2.61i 185 . 2 ( +𝑣𝑈) = (1st ‘(1st𝑈))
171, 16eqtri 2821 1 𝐺 = (1st ‘(1st𝑈))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243   ∘ ccom 5524  ⟶wf 6323  –onto→wfo 6325  ‘cfv 6327  1st c1st 7676   +𝑣 cpv 28382 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 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fo 6333  df-fv 6335  df-1st 7678  df-va 28392 This theorem is referenced by:  nvvop  28406  nvablo  28413  nvsf  28416  nvscl  28423  nvsid  28424  nvsass  28425  nvdi  28427  nvdir  28428  nv2  28429  nv0  28434  nvsz  28435  nvinv  28436  cnnvg  28475  phop  28615  ip0i  28622  ipdirilem  28626  h2hva  28771  hhssva  29054  hhshsslem1  29064
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