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Theorem isnvc2 22426
 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 22422 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2 nlmlmod 22405 . . . 4 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3 isnvc2.1 . . . . . 6 𝐹 = (Scalar‘𝑊)
43islvec 19036 . . . . 5 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
54baib 943 . . . 4 (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
62, 5syl 17 . . 3 (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
76pm5.32i 668 . 2 ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
81, 7bitri 264 1 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ‘cfv 5852  Scalarcsca 15876  DivRingcdr 18679  LModclmod 18795  LVecclvec 19034  NrmModcnlm 22308  NrmVeccnvc 22309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-lvec 19035  df-nlm 22314  df-nvc 22315 This theorem is referenced by:  lssnvc  22429  srabn  23079
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