Theorem isnvc2 24735

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

Step	Hyp	Ref	Expression
1		isnvc 24731	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2		nlmlmod 24714	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3		isnvc2.1	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
4	3	islvec 21120	. . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
5	4	baib 535	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
6	2, 5	syl 17	. . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
7	6	pm5.32i 574	. 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
8	1, 7	bitri 275	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ‘cfv 6562  Scalarcsca 17300  DivRingcdr 20745  LModclmod 20874  LVecclvec 21118  NrmModcnlm 24608  NrmVeccnvc 24609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-nul 5311

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-lvec 21119  df-nlm 24614  df-nvc 24615

This theorem is referenced by:  lssnvc  24738  srabn  25407