Theorem nvablo 30706

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 2737	. . 3 ⊢ (1st ‘𝑈) = (1st ‘𝑈)
2	1	nvvc 30705	. 2 ⊢ (𝑈 ∈ NrmCVec → (1st ‘𝑈) ∈ CVecOLD)
3		nvabl.1	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4	3	vafval 30693	. . 3 ⊢ 𝐺 = (1st ‘(1st ‘𝑈))
5	4	vcablo 30659	. 2 ⊢ ((1st ‘𝑈) ∈ CVecOLD → 𝐺 ∈ AbelOp)
6	2, 5	syl 17	1 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6494  1^st c1st 7935  AbelOpcablo 30634  CVec_OLDcvc 30648  NrmCVeccnv 30674   +_𝑣 cpv 30675

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-1st 7937  df-2nd 7938  df-vc 30649  df-nv 30682  df-va 30685  df-ba 30686  df-sm 30687  df-0v 30688  df-nmcv 30690

This theorem is referenced by:  nvgrp  30707  nvcom  30711  nvadd32  30713  nvadd4  30715  nvnnncan1  30737  nvaddsub  30745