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Theorem lmhmlvec2 33158
Description: A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.)
Assertion
Ref Expression
lmhmlvec2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)

Proof of Theorem lmhmlvec2
StepHypRef Expression
1 lmhmlmod2 20876 . . 3 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
21adantl 481 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod)
3 eqid 2731 . . . . 5 (Scalar‘𝑉) = (Scalar‘𝑉)
4 eqid 2731 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
53, 4lmhmsca 20874 . . . 4 (𝐹 ∈ (𝑉 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑉))
65adantl 481 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) = (Scalar‘𝑉))
73lvecdrng 20949 . . . 4 (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing)
87adantr 480 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing)
96, 8eqeltrd 2832 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing)
104islvec 20948 . 2 (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing))
112, 9, 10sylanbrc 582 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  cfv 6543  (class class class)co 7412  Scalarcsca 17207  DivRingcdr 20583  LModclmod 20702   LMHom clmhm 20863  LVecclvec 20946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-lmhm 20866  df-lvec 20947
This theorem is referenced by:  imlmhm  33160  dimkerim  33166  algextdeglem8  33235
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