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Theorem 0vfval 27589
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 3243 . 2 (𝑈𝑉𝑈 ∈ V)
2 fo1st 7230 . . . . . . 7 1st :V–onto→V
3 fofn 6155 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . 6 1st Fn V
5 ssv 3658 . . . . . 6 ran 1st ⊆ V
6 fnco 6037 . . . . . 6 ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
74, 4, 5, 6mp3an 1464 . . . . 5 (1st ∘ 1st ) Fn V
8 df-va 27578 . . . . . 6 +𝑣 = (1st ∘ 1st )
98fneq1i 6023 . . . . 5 ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 221 . . . 4 +𝑣 Fn V
11 fvco2 6312 . . . 4 (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
1210, 11mpan 706 . . 3 (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
13 0vfval.5 . . . 4 𝑍 = (0vec𝑈)
14 df-0v 27581 . . . . 5 0vec = (GId ∘ +𝑣 )
1514fveq1i 6230 . . . 4 (0vec𝑈) = ((GId ∘ +𝑣 )‘𝑈)
1613, 15eqtri 2673 . . 3 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17 0vfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1817fveq2i 6232 . . 3 (GId‘𝐺) = (GId‘( +𝑣𝑈))
1912, 16, 183eqtr4g 2710 . 2 (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
201, 19syl 17 1 (𝑈𝑉𝑍 = (GId‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  wss 3607  ran crn 5144  ccom 5147   Fn wfn 5921  ontowfo 5924  cfv 5926  1st c1st 7208  GIdcgi 27472   +𝑣 cpv 27568  0veccn0v 27571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-va 27578  df-0v 27581
This theorem is referenced by:  nvi  27597  nvzcl  27617  nv0rid  27618  nv0lid  27619  nv0  27620  nvsz  27621  nvrinv  27634  nvlinv  27635  hh0v  28153
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