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| Description: The vector addition operation of a normed complex vector space is a group. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| nvabl.1 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) | 
| Ref | Expression | 
|---|---|
| nvgrp | ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nvabl.1 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
| 2 | 1 | nvablo 30635 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ AbelOp) | 
| 3 | ablogrpo 30566 | . 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
| 4 | 2, 3 | syl 17 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 GrpOpcgr 30508 AbelOpcablo 30563 NrmCVeccnv 30603 +𝑣 cpv 30604 | 
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-1st 8014 df-2nd 8015 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-nmcv 30619 | 
| This theorem is referenced by: nvgf 30637 nvgcl 30639 nvass 30641 nvrcan 30643 nvzcl 30653 nv0rid 30654 nv0lid 30655 nvinvfval 30659 nvmval 30661 nvmfval 30663 nvnegneg 30668 nvrinv 30670 nvlinv 30671 hhshsslem1 31286 | 
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