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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 24716 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
| 2 | nlmlmod 24699 | . . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
| 3 | isnvc2.1 | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 4 | 3 | islvec 21103 | . . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
| 5 | 4 | baib 535 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
| 6 | 2, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
| 7 | 6 | pm5.32i 574 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
| 8 | 1, 7 | bitri 275 | 1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Scalarcsca 17300 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 NrmModcnlm 24593 NrmVeccnvc 24594 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lvec 21102 df-nlm 24599 df-nvc 24600 |
| This theorem is referenced by: lssnvc 24723 srabn 25394 |
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