| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nlmngp | Structured version Visualization version GIF version | ||
| Description: A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
| Ref | Expression |
|---|---|
| nlmngp | ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
| 3 | eqid 2737 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | eqid 2737 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 5 | eqid 2737 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 6 | eqid 2737 | . . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊)) | |
| 7 | 1, 2, 3, 4, 5, 6 | isnlm 24696 | . . 3 ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝑊))∀𝑦 ∈ (Base‘𝑊)((norm‘𝑊)‘(𝑥( ·𝑠 ‘𝑊)𝑦)) = (((norm‘(Scalar‘𝑊))‘𝑥) · ((norm‘𝑊)‘𝑦)))) |
| 8 | 7 | simplbi 497 | . 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing)) |
| 9 | 8 | simp1d 1143 | 1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 · cmul 11160 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 LModclmod 20858 normcnm 24589 NrmGrpcngp 24590 NrmRingcnrg 24592 NrmModcnlm 24593 |
| 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-ext 2708 ax-nul 5306 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 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-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-nlm 24599 |
| This theorem is referenced by: nlmdsdi 24702 nlmdsdir 24703 nlmmul0or 24704 nlmvscnlem2 24706 nlmvscnlem1 24707 nlmvscn 24708 nlmtlm 24715 lssnlm 24722 ngpocelbl 24725 isnmhm2 24773 idnmhm 24775 0nmhm 24776 nmoleub2lem 25147 nmoleub2lem3 25148 nmoleub2lem2 25149 nmoleub3 25152 nmhmcn 25153 ncvsm1 25188 ncvsdif 25189 ncvspi 25190 ncvs1 25191 ncvspds 25195 cphngp 25207 ipcnlem2 25278 ipcnlem1 25279 csscld 25283 bnngp 25376 cssbn 25409 |
| Copyright terms: Public domain | W3C validator |