Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nlmngp2 | Structured version Visualization version Unicode version |
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | Scalar |
Ref | Expression |
---|---|
nlmngp2 | NrmMod NrmGrp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmnrg.1 | . . 3 Scalar | |
2 | 1 | nlmnrg 22483 | . 2 NrmMod NrmRing |
3 | nrgngp 22466 | . 2 NrmRing NrmGrp | |
4 | 2, 3 | syl 17 | 1 NrmMod NrmGrp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 cfv 5888 Scalarcsca 15944 NrmGrpcngp 22382 NrmRingcnrg 22384 NrmModcnlm 22385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-nrg 22390 df-nlm 22391 |
This theorem is referenced by: nlmdsdir 22486 nlmmul0or 22487 nlmvscnlem2 22489 nlmvscnlem1 22490 nlmvscn 22491 |
Copyright terms: Public domain | W3C validator |