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Theorem 0vfval 28310
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 3510 . 2 (𝑈𝑉𝑈 ∈ V)
2 fo1st 7698 . . . . . . 7 1st :V–onto→V
3 fofn 6585 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . 6 1st Fn V
5 ssv 3988 . . . . . 6 ran 1st ⊆ V
6 fnco 6458 . . . . . 6 ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
74, 4, 5, 6mp3an 1452 . . . . 5 (1st ∘ 1st ) Fn V
8 df-va 28299 . . . . . 6 +𝑣 = (1st ∘ 1st )
98fneq1i 6443 . . . . 5 ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 232 . . . 4 +𝑣 Fn V
11 fvco2 6751 . . . 4 (( +𝑣 Fn V ∧ 𝑈 ∈ V) → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
1210, 11mpan 686 . . 3 (𝑈 ∈ V → ((GId ∘ +𝑣 )‘𝑈) = (GId‘( +𝑣𝑈)))
13 0vfval.5 . . . 4 𝑍 = (0vec𝑈)
14 df-0v 28302 . . . . 5 0vec = (GId ∘ +𝑣 )
1514fveq1i 6664 . . . 4 (0vec𝑈) = ((GId ∘ +𝑣 )‘𝑈)
1613, 15eqtri 2841 . . 3 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17 0vfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1817fveq2i 6666 . . 3 (GId‘𝐺) = (GId‘( +𝑣𝑈))
1912, 16, 183eqtr4g 2878 . 2 (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
201, 19syl 17 1 (𝑈𝑉𝑍 = (GId‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  ran crn 5549  ccom 5552   Fn wfn 6343  ontowfo 6346  cfv 6348  1st c1st 7676  GIdcgi 28194   +𝑣 cpv 28289  0veccn0v 28292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-va 28299  df-0v 28302
This theorem is referenced by:  nvi  28318  nvzcl  28338  nv0rid  28339  nv0lid  28340  nv0  28341  nvsz  28342  nvrinv  28355  nvlinv  28356  hh0v  28872
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