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Mirrors > Home > MPE Home > Th. List > isnvc2 | Structured version Visualization version GIF version |
Description: A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
isnvc2.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
isnvc2 | ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvc 22907 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | nlmlmod 22890 | . . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
3 | isnvc2.1 | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | islvec 19499 | . . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
5 | 4 | baib 531 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
7 | 6 | pm5.32i 570 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
8 | 1, 7 | bitri 267 | 1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 Scalarcsca 16341 DivRingcdr 19139 LModclmod 19255 LVecclvec 19497 NrmModcnlm 22793 NrmVeccnvc 22794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5025 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-iota 6099 df-fv 6143 df-ov 6925 df-lvec 19498 df-nlm 22799 df-nvc 22800 |
This theorem is referenced by: lssnvc 22914 srabn 23566 |
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