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Theorem nvablo 29900
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 2733 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 29899 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvabl.1 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 29887 . . 3 𝐺 = (1st ‘(1st𝑈))
54vcablo 29853 . 2 ((1st𝑈) ∈ CVecOLD𝐺 ∈ AbelOp)
62, 5syl 17 1 (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cfv 6544  1st c1st 7973  AbelOpcablo 29828  CVecOLDcvc 29842  NrmCVeccnv 29868   +𝑣 cpv 29869
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29843  df-nv 29876  df-va 29879  df-ba 29880  df-sm 29881  df-0v 29882  df-nmcv 29884
This theorem is referenced by:  nvgrp  29901  nvcom  29905  nvadd32  29907  nvadd4  29909  nvnnncan1  29931  nvaddsub  29939
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