Theorem isnvc2 22723

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 22719	. 2 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
2		nlmlmod 22702	. . . 4 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
3		isnvc2.1	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
4	3	islvec 19317	. . . . 5 ⊢ (𝑊 ∈ LVec ↔ (𝑊 ∈ LMod ∧ 𝐹 ∈ DivRing))
5	4	baib 525	. . . 4 ⊢ (𝑊 ∈ LMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
6	2, 5	syl 17	. . 3 ⊢ (𝑊 ∈ NrmMod → (𝑊 ∈ LVec ↔ 𝐹 ∈ DivRing))
7	6	pm5.32i 564	. 2 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec) ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))
8	1, 7	bitri 264	1 ⊢ (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Syntax hints:   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ‘cfv 6030  Scalarcsca 16152  DivRingcdr 18957  LModclmod 19073  LVecclvec 19315  NrmModcnlm 22605  NrmVeccnvc 22606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-lvec 19316  df-nlm 22611  df-nvc 22612

This theorem is referenced by:  lssnvc  22726  srabn  23375