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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 24532 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | nlmlmod 24515 | . . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
3 | isnvc2.1 | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | islvec 20948 | . . . . 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 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 Scalarcsca 17207 DivRingcdr 20583 LModclmod 20702 LVecclvec 20946 NrmModcnlm 24409 NrmVeccnvc 24410 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-lvec 20947 df-nlm 24415 df-nvc 24416 |
This theorem is referenced by: lssnvc 24539 srabn 25208 |
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