MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvablo Structured version   Visualization version   GIF version

Theorem nvablo 28964
Description: The vector addition operation of a normed complex vector space is an Abelian group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvabl.1 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
nvablo (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Proof of Theorem nvablo
StepHypRef Expression
1 eqid 2738 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28963 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvabl.1 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 28951 . . 3 𝐺 = (1st ‘(1st𝑈))
54vcablo 28917 . 2 ((1st𝑈) ∈ CVecOLD𝐺 ∈ AbelOp)
62, 5syl 17 1 (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  cfv 6427  1st c1st 7819  AbelOpcablo 28892  CVecOLDcvc 28906  NrmCVeccnv 28932   +𝑣 cpv 28933
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5222  ax-nul 5229  ax-pr 5351  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3432  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5485  df-xp 5591  df-rel 5592  df-cnv 5593  df-co 5594  df-dm 5595  df-rn 5596  df-res 5597  df-ima 5598  df-iota 6385  df-fun 6429  df-fn 6430  df-f 6431  df-f1 6432  df-fo 6433  df-f1o 6434  df-fv 6435  df-ov 7271  df-oprab 7272  df-1st 7821  df-2nd 7822  df-vc 28907  df-nv 28940  df-va 28943  df-ba 28944  df-sm 28945  df-0v 28946  df-nmcv 28948
This theorem is referenced by:  nvgrp  28965  nvcom  28969  nvadd32  28971  nvadd4  28973  nvnnncan1  28995  nvaddsub  29003
  Copyright terms: Public domain W3C validator