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Theorem nlmngp 24638

Description: A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)

**Assertion**
Ref	Expression
nlmngp	⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Proof of Theorem nlmngp Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2725	. . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2725	. . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊)
3		eqid 2725	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
4		eqid 2725	. . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
5		eqid 2725	. . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
6		eqid 2725	. . . 4 ⊢ (norm‘(Scalar‘𝑊)) = (norm‘(Scalar‘𝑊))
7	1, 2, 3, 4, 5, 6	isnlm 24636	. . 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 496	. 2 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ (Scalar‘𝑊) ∈ NrmRing))
9	8	simp1d 1139	1 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ‘cfv 6549 (class class class)co 7419 · cmul 11145 Basecbs 17183 Scalarcsca 17239 ·_𝑠 cvsca 17240 LModclmod 20755 normcnm 24529 NrmGrpcngp 24530 NrmRingcnrg 24532 NrmModcnlm 24533

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5307

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-nlm 24539

This theorem is referenced by: nlmdsdi 24642 nlmdsdir 24643 nlmmul0or 24644 nlmvscnlem2 24646 nlmvscnlem1 24647 nlmvscn 24648 nlmtlm 24655 lssnlm 24662 ngpocelbl 24665 isnmhm2 24713 idnmhm 24715 0nmhm 24716 nmoleub2lem 25085 nmoleub2lem3 25086 nmoleub2lem2 25087 nmoleub3 25090 nmhmcn 25091 ncvsm1 25126 ncvsdif 25127 ncvspi 25128 ncvs1 25129 ncvspds 25133 cphngp 25145 ipcnlem2 25216 ipcnlem1 25217 csscld 25221 bnngp 25314 cssbn 25347

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