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Theorem 0vfval 27445
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 3210 . 2 (𝑈𝑉𝑈 ∈ V)
2 fo1st 7185 . . . . . . 7 1st :V–onto→V
3 fofn 6115 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
42, 3ax-mp 5 . . . . . 6 1st Fn V
5 ssv 3623 . . . . . 6 ran 1st ⊆ V
6 fnco 5997 . . . . . 6 ((1st Fn V ∧ 1st Fn V ∧ ran 1st ⊆ V) → (1st ∘ 1st ) Fn V)
74, 4, 5, 6mp3an 1423 . . . . 5 (1st ∘ 1st ) Fn V
8 df-va 27434 . . . . . 6 +𝑣 = (1st ∘ 1st )
98fneq1i 5983 . . . . 5 ( +𝑣 Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 221 . . . 4 +𝑣 Fn V
11 fvco2 6271 . . . 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 27437 . . . . 5 0vec = (GId ∘ +𝑣 )
1514fveq1i 6190 . . . 4 (0vec𝑈) = ((GId ∘ +𝑣 )‘𝑈)
1613, 15eqtri 2643 . . 3 𝑍 = ((GId ∘ +𝑣 )‘𝑈)
17 0vfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1817fveq2i 6192 . . 3 (GId‘𝐺) = (GId‘( +𝑣𝑈))
1912, 16, 183eqtr4g 2680 . 2 (𝑈 ∈ V → 𝑍 = (GId‘𝐺))
201, 19syl 17 1 (𝑈𝑉𝑍 = (GId‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  ran crn 5113  ccom 5116   Fn wfn 5881  ontowfo 5884  cfv 5886  1st c1st 7163  GIdcgi 27328   +𝑣 cpv 27424  0veccn0v 27427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-1st 7165  df-va 27434  df-0v 27437
This theorem is referenced by:  nvi  27453  nvzcl  27473  nv0rid  27474  nv0lid  27475  nv0  27476  nvsz  27477  nvrinv  27490  nvlinv  27491  hh0v  28009
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