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Theorem nvgrp 28378
Description: The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvabl.1 𝐺 = ( +𝑣𝑈)
Assertion
Ref Expression
nvgrp (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)

Proof of Theorem nvgrp
StepHypRef Expression
1 nvabl.1 . . 3 𝐺 = ( +𝑣𝑈)
21nvablo 28377 . 2 (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp)
3 ablogrpo 28308 . 2 (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp)
42, 3syl 17 1 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cfv 6328  GrpOpcgr 28250  AbelOpcablo 28305  NrmCVeccnv 28345   +𝑣 cpv 28346
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-1st 7664  df-2nd 7665  df-ablo 28306  df-vc 28320  df-nv 28353  df-va 28356  df-ba 28357  df-sm 28358  df-0v 28359  df-nmcv 28361
This theorem is referenced by:  nvgf  28379  nvgcl  28381  nvass  28383  nvrcan  28385  nvzcl  28395  nv0rid  28396  nv0lid  28397  nvinvfval  28401  nvmval  28403  nvmfval  28405  nvnegneg  28410  nvrinv  28412  nvlinv  28413  hhshsslem1  29028
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