Theorem isnvc2 23311

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 23307	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2		nlmlmod 23290	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3		isnvc2.1	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
4	3	islvec 19879	. . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
5	4	baib 538	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
6	2, 5	syl 17	. . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
7	6	pm5.32i 577	. 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
8	1, 7	bitri 277	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ‘cfv 6358  Scalarcsca 16571  DivRingcdr 19505  LModclmod 19637  LVecclvec 19877  NrmModcnlm 23193  NrmVeccnvc 23194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-nul 5213

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-iota 6317  df-fv 6366  df-ov 7162  df-lvec 19878  df-nlm 23199  df-nvc 23200

This theorem is referenced by:  lssnvc  23314  srabn  23966