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Theorem 0vfval 26845
 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 3185 . 2 (𝑈𝑉𝑈 ∈ V)
2 fo1st 7079 . . . . . . 7 1st :V–onto→V
3 fofn 6030 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . 6 1st Fn V
5 ssv 3588 . . . . . 6 ran 1st ⊆ V
6 fnco 5913 . . . . . 6 ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
74, 4, 5, 6mp3an 1416 . . . . 5 (1st ∘ 1st ) Fn V
8 df-va 26834 . . . . . 6 +𝑣 = (1st ∘ 1st )
98fneq1i 5899 . . . . 5 ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 220 . . . 4 +𝑣 Fn V
11 fvco2 6183 . . . 4 (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
1210, 11mpan 702 . . 3 (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
13 0vfval.5 . . . 4 𝑍 = (0vec𝑈)
14 df-0v 26837 . . . . 5 0vec = (GId ∘ +𝑣 )
1514fveq1i 6104 . . . 4 (0vec𝑈) = ((GId ∘ +𝑣 )‘𝑈)
1613, 15eqtri 2632 . . 3 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17 0vfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1817fveq2i 6106 . . 3 (GId‘𝐺) = (GId‘( +𝑣𝑈))
1912, 16, 183eqtr4g 2669 . 2 (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
201, 19syl 17 1 (𝑈𝑉𝑍 = (GId‘𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ran crn 5039   ∘ ccom 5042   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  1st c1st 7057  GIdcgi 26728   +𝑣 cpv 26824  0veccn0v 26827 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-va 26834  df-0v 26837 This theorem is referenced by:  nvi  26853  nvzcl  26873  nv0rid  26874  nv0lid  26875  nv0  26876  nvsz  26877  nvrinv  26890  nvlinv  26891  hh0v  27409
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