Theorem nvablo 28978

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 2738	. . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2	1	nvvc 28977	. 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD)
3		nvabl.1	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	3	vafval 28965	. . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
5	4	vcablo 28931	. 2 ⊢ ((1st ‘𝑈) ∈ CVecOLD → 𝐺 ∈ AbelOp)
6	2, 5	syl 17	1 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  1^st c1st 7829  AbelOpcablo 28906  CVec_OLDcvc 28920  NrmCVeccnv 28946   +_𝑣 cpv 28947

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 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

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 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-1st 7831  df-2nd 7832  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962

This theorem is referenced by:  nvgrp  28979  nvcom  28983  nvadd32  28985  nvadd4  28987  nvnnncan1  29009  nvaddsub  29017