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Theorem isnvc2 24086
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 = (Scalarβ€˜π‘Š)
Assertion
Ref Expression
isnvc2 (π‘Š ∈ NrmVec ↔ (π‘Š ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Proof of Theorem isnvc2
StepHypRef Expression
1 isnvc 24082 . 2 (π‘Š ∈ NrmVec ↔ (π‘Š ∈ NrmMod ∧ π‘Š ∈ LVec))
2 nlmlmod 24065 . . . 4 (π‘Š ∈ NrmMod β†’ π‘Š ∈ LMod)
3 isnvc2.1 . . . . . 6 𝐹 = (Scalarβ€˜π‘Š)
43islvec 20609 . . . . 5 (π‘Š ∈ LVec ↔ (π‘Š ∈ LMod ∧ 𝐹 ∈ DivRing))
54baib 537 . . . 4 (π‘Š ∈ LMod β†’ (π‘Š ∈ LVec ↔ 𝐹 ∈ DivRing))
62, 5syl 17 . . 3 (π‘Š ∈ NrmMod β†’ (π‘Š ∈ LVec ↔ 𝐹 ∈ DivRing))
76pm5.32i 576 . 2 ((π‘Š ∈ NrmMod ∧ π‘Š ∈ LVec) ↔ (π‘Š ∈ NrmMod ∧ 𝐹 ∈ DivRing))
81, 7bitri 275 1 (π‘Š ∈ NrmVec ↔ (π‘Š ∈ NrmMod ∧ 𝐹 ∈ DivRing))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  Scalarcsca 17144  DivRingcdr 20219  LModclmod 20365  LVecclvec 20607  NrmModcnlm 23959  NrmVeccnvc 23960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-iota 6452  df-fv 6508  df-ov 7364  df-lvec 20608  df-nlm 23965  df-nvc 23966
This theorem is referenced by:  lssnvc  24089  srabn  24747
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