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Theorem nvablo 30124
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 2732 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 30123 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 nvabl.1 . . . 4 𝐺 = ( +𝑣𝑈)
43vafval 30111 . . 3 𝐺 = (1st ‘(1st𝑈))
54vcablo 30077 . 2 ((1st𝑈) ∈ CVecOLD𝐺 ∈ AbelOp)
62, 5syl 17 1 (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cfv 6543  1st c1st 7975  AbelOpcablo 30052  CVecOLDcvc 30066  NrmCVeccnv 30092   +𝑣 cpv 30093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-1st 7977  df-2nd 7978  df-vc 30067  df-nv 30100  df-va 30103  df-ba 30104  df-sm 30105  df-0v 30106  df-nmcv 30108
This theorem is referenced by:  nvgrp  30125  nvcom  30129  nvadd32  30131  nvadd4  30133  nvnnncan1  30155  nvaddsub  30163
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