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Theorem isnvc2 23309
Description: A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypothesis
Ref Expression
isnvc2.1 𝐹 = (Scalar‘𝑊)
Assertion
Ref Expression
isnvc2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Proof of Theorem isnvc2
StepHypRef Expression
1 isnvc 23305 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2 nlmlmod 23288 . . . 4 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3 isnvc2.1 . . . . . 6 𝐹 = (Scalar‘𝑊)
43islvec 19873 . . . . 5 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
54baib 539 . . . 4 (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
62, 5syl 17 . . 3 (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
76pm5.32i 578 . 2 ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
81, 7bitri 278 1 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  cfv 6328  Scalarcsca 16564  DivRingcdr 19499  LModclmod 19631  LVecclvec 19871  NrmModcnlm 23191  NrmVeccnvc 23192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142  df-lvec 19872  df-nlm 23197  df-nvc 23198
This theorem is referenced by:  lssnvc  23312  srabn  23968
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