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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 23306 | . 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec)) | |
2 | nlmlmod 23289 | . . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
3 | isnvc2.1 | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
4 | 3 | islvec 19878 | . . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing)) |
5 | 4 | baib 538 | . . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
6 | 2, 5 | syl 17 | . . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing)) |
7 | 6 | pm5.32i 577 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
8 | 1, 7 | bitri 277 | 1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Scalarcsca 16570 DivRingcdr 19504 LModclmod 19636 LVecclvec 19876 NrmModcnlm 23192 NrmVeccnvc 23193 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-lvec 19877 df-nlm 23198 df-nvc 23199 |
This theorem is referenced by: lssnvc 23313 srabn 23965 |
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