Theorem nvablo 30645

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 = ( +𝑣 ‘𝑈)

Assertion

Ref	Expression
nvablo	⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Proof of Theorem nvablo

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2	1	nvvc 30644	. 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD)
3		nvabl.1	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	3	vafval 30632	. . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
5	4	vcablo 30598	. 2 ⊢ ((1st ‘𝑈) ∈ CVecOLD → 𝐺 ∈ AbelOp)
6	2, 5	syl 17	1 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  1^st c1st 8011  AbelOpcablo 30573  CVec_OLDcvc 30587  NrmCVeccnv 30613   +_𝑣 cpv 30614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629

This theorem is referenced by:  nvgrp  30646  nvcom  30650  nvadd32  30652  nvadd4  30654  nvnnncan1  30676  nvaddsub  30684