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Mirrors > Home > MPE Home > Th. List > bnngp | Structured version Visualization version GIF version |
Description: A Banach space is a normed group. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
bnngp | ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnlm 24495 | . 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod) | |
2 | nlmngp 23831 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 NrmGrpcngp 23723 NrmModcnlm 23726 Bancbn 24487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-nul 5234 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-iota 6389 df-fv 6439 df-ov 7272 df-nlm 23732 df-nvc 23733 df-bn 24490 |
This theorem is referenced by: (None) |
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