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Theorem 0vfval 29590
Description: Value of the function for the zero vector on a normed complex vector space. (Contributed by NM, 24-Apr-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
0vfval.2 𝐺 = ( +𝑣𝑈)
0vfval.5 𝑍 = (0vec𝑈)
Assertion
Ref Expression
0vfval (𝑈𝑉𝑍 = (GId‘𝐺))

Proof of Theorem 0vfval
StepHypRef Expression
1 elex 3462 . 2 (𝑈𝑉𝑈 ∈ V)
2 fo1st 7942 . . . . . . 7 1st :V–onto→V
3 fofn 6759 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . 6 1st Fn V
5 ssv 3969 . . . . . 6 ran 1st ⊆ V
6 fnco 6619 . . . . . 6 ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
74, 4, 5, 6mp3an 1462 . . . . 5 (1st ∘ 1st ) Fn V
8 df-va 29579 . . . . . 6 +𝑣 = (1st ∘ 1st )
98fneq1i 6600 . . . . 5 ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 230 . . . 4 +𝑣 Fn V
11 fvco2 6939 . . . 4 (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
1210, 11mpan 689 . . 3 (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
13 0vfval.5 . . . 4 𝑍 = (0vec𝑈)
14 df-0v 29582 . . . . 5 0vec = (GId ∘ +𝑣 )
1514fveq1i 6844 . . . 4 (0vec𝑈) = ((GId ∘ +𝑣 )‘𝑈)
1613, 15eqtri 2761 . . 3 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17 0vfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1817fveq2i 6846 . . 3 (GId‘𝐺) = (GId‘( +𝑣𝑈))
1912, 16, 183eqtr4g 2798 . 2 (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
201, 19syl 17 1 (𝑈𝑉𝑍 = (GId‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  ran crn 5635  ccom 5638   Fn wfn 6492  ontowfo 6495  cfv 6497  1st c1st 7920  GIdcgi 29474   +𝑣 cpv 29569  0veccn0v 29572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-va 29579  df-0v 29582
This theorem is referenced by:  nvi  29598  nvzcl  29618  nv0rid  29619  nv0lid  29620  nv0  29621  nvsz  29622  nvrinv  29635  nvlinv  29636  hh0v  30152
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