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Theorem lmhmlvec2 31038
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 19790 . . 3 (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
21adantl 485 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod)
3 eqid 2824 . . . . 5 (Scalar‘𝑉) = (Scalar‘𝑉)
4 eqid 2824 . . . . 5 (Scalar‘𝑈) = (Scalar‘𝑈)
53, 4lmhmsca 19788 . . . 4 (𝐹 ∈ (𝑉 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑉))
65adantl 485 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) = (Scalar‘𝑉))
73lvecdrng 19863 . . . 4 (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing)
87adantr 484 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing)
96, 8eqeltrd 2916 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing)
104islvec 19862 . 2 (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing))
112, 9, 10sylanbrc 586 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cfv 6336  (class class class)co 7138  Scalarcsca 16557  DivRingcdr 19488  LModclmod 19620   LMHom clmhm 19777  LVecclvec 19860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-opab 5110  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-iota 6295  df-fun 6338  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-lmhm 19780  df-lvec 19861
This theorem is referenced by:  imlmhm  31040  dimkerim  31044
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