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Theorem vafval 27914
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 27906 . . . . 5 +𝑣 = (1st ∘ 1st )
32fveq1i 6376 . . . 4 ( +𝑣𝑈) = ((1st ∘ 1st )‘𝑈)
4 fo1st 7386 . . . . . 6 1st :V–onto→V
5 fof 6298 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fvco3 6464 . . . . 5 ((1st :V⟶V ∧ 𝑈 ∈ V) → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
86, 7mpan 681 . . . 4 (𝑈 ∈ V → ((1st ∘ 1st )‘𝑈) = (1st ‘(1st𝑈)))
93, 8syl5eq 2811 . . 3 (𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
10 fvprc 6368 . . . 4 𝑈 ∈ V → ( +𝑣𝑈) = ∅)
11 fvprc 6368 . . . . . 6 𝑈 ∈ V → (1st𝑈) = ∅)
1211fveq2d 6379 . . . . 5 𝑈 ∈ V → (1st ‘(1st𝑈)) = (1st ‘∅))
13 1st0 7372 . . . . 5 (1st ‘∅) = ∅
1412, 13syl6req 2816 . . . 4 𝑈 ∈ V → ∅ = (1st ‘(1st𝑈)))
1510, 14eqtrd 2799 . . 3 𝑈 ∈ V → ( +𝑣𝑈) = (1st ‘(1st𝑈)))
169, 15pm2.61i 176 . 2 ( +𝑣𝑈) = (1st ‘(1st𝑈))
171, 16eqtri 2787 1 𝐺 = (1st ‘(1st𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  ccom 5281  wf 6064  ontowfo 6066  cfv 6068  1st c1st 7364   +𝑣 cpv 27896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-va 27906
This theorem is referenced by:  nvvop  27920  nvablo  27927  nvsf  27930  nvscl  27937  nvsid  27938  nvsass  27939  nvdi  27941  nvdir  27942  nv2  27943  nv0  27948  nvsz  27949  nvinv  27950  cnnvg  27989  phop  28129  ip0i  28136  ipdirilem  28140  h2hva  28287  hhssva  28570  hhshsslem1  28580
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