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Theorem vafval 28063
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 28055 . . . . 5 +𝑣 = (1st ∘ 1st )
32fveq1i 6542 . . . 4 ( +𝑣𝑈) = ((1st ∘ 1st )‘𝑈)
4 fo1st 7568 . . . . . 6 1st :V–onto→V
5 fof 6461 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6630 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
86, 7mpan 686 . . . 4 (𝑈 ∈ V → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
93, 8syl5eq 2842 . . 3 (𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
10 fvprc 6534 . . . 4 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
11 fvprc 6534 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6545 . . . . 5 𝑈 ∈ V → (1st ‘(1st𝑈)) = (1st ‘∅))
13 1st0 7554 . . . . 5 (1st ‘∅) = ∅
1412, 13syl6req 2847 . . . 4 𝑈 ∈ V → ∅ = (1st ‘(1st𝑈)))
1510, 14eqtrd 2830 . . 3 𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
169, 15pm2.61i 183 . 2 ( +𝑣𝑈) = (1st ‘(1st𝑈))
171, 16eqtri 2818 1 𝐺 = (1st ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1522  wcel 2080  Vcvv 3436  c0 4213  ccom 5450  wf 6224  ontowfo 6226  cfv 6228  1st c1st 7546   +𝑣 cpv 28045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fo 6234  df-fv 6236  df-1st 7548  df-va 28055
This theorem is referenced by:  nvvop  28069  nvablo  28076  nvsf  28079  nvscl  28086  nvsid  28087  nvsass  28088  nvdi  28090  nvdir  28091  nv2  28092  nv0  28097  nvsz  28098  nvinv  28099  cnnvg  28138  phop  28278  ip0i  28285  ipdirilem  28289  h2hva  28434  hhssva  28717  hhshsslem1  28727
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