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Theorem isnvc2 22911
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 22907 . 2 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2 nlmlmod 22890 . . . 4 (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3 isnvc2.1 . . . . . 6 𝐹 = (Scalar‘𝑊)
43islvec 19499 . . . . 5 (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
54baib 531 . . . 4 (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
62, 5syl 17 . . 3 (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
76pm5.32i 570 . 2 ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
81, 7bitri 267 1 (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  cfv 6135  Scalarcsca 16341  DivRingcdr 19139  LModclmod 19255  LVecclvec 19497  NrmModcnlm 22793  NrmVeccnvc 22794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-lvec 19498  df-nlm 22799  df-nvc 22800
This theorem is referenced by:  lssnvc  22914  srabn  23566
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