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Mirrors > Home > MPE Home > Th. List > nlmngp2 | Structured version Visualization version GIF version |
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
nlmnrg.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
nlmngp2 | ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlmnrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | 1 | nlmnrg 23290 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing) |
3 | nrgngp 23273 | . 2 ⊢ (𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Scalarcsca 16570 NrmGrpcngp 23189 NrmRingcnrg 23191 NrmModcnlm 23192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-nrg 23197 df-nlm 23198 |
This theorem is referenced by: nlmdsdir 23293 nlmmul0or 23294 nlmvscnlem2 23296 nlmvscnlem1 23297 nlmvscn 23298 |
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