Theorem nvablo 30595

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 2729	. . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2	1	nvvc 30594	. 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD)
3		nvabl.1	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	3	vafval 30582	. . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
5	4	vcablo 30548	. 2 ⊢ ((1st ‘𝑈) ∈ CVecOLD → 𝐺 ∈ AbelOp)
6	2, 5	syl 17	1 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6499  1^st c1st 7945  AbelOpcablo 30523  CVec_OLDcvc 30537  NrmCVeccnv 30563   +_𝑣 cpv 30564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-1st 7947  df-2nd 7948  df-vc 30538  df-nv 30571  df-va 30574  df-ba 30575  df-sm 30576  df-0v 30577  df-nmcv 30579

This theorem is referenced by:  nvgrp  30596  nvcom  30600  nvadd32  30602  nvadd4  30604  nvnnncan1  30626  nvaddsub  30634