Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > bnlmod | Structured version Visualization version GIF version |
Description: A Banach space is a left module. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
bnlmod | ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnnlm 24106 | . 2 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ NrmMod) | |
2 | nlmlmod 23444 | . 2 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ Ban → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 LModclmod 19766 NrmModcnlm 23346 Bancbn 24098 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-ov 7186 df-nlm 23352 df-nvc 23353 df-bn 24101 |
This theorem is referenced by: sitgclbn 31893 |
Copyright terms: Public domain | W3C validator |