Theorem lmhmlvec2 33763

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

Step	Hyp	Ref	Expression
1		lmhmlmod2 21027	. . 3 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod)
2	1	adantl 481	. 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod)
3		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉)
4		eqid 2736	. . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
5	3, 4	lmhmsca 21025	. . . 4 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑉))
6	5	adantl 481	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) = (Scalar‘𝑉))
7	3	lvecdrng 21100	. . . 4 ⊢ (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing)
8	7	adantr 480	. . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing)
9	6, 8	eqeltrd 2836	. 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing)
10	4	islvec 21099	. 2 ⊢ (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing))
11	2, 9, 10	sylanbrc 584	1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Scalarcsca 17223  DivRingcdr 20706  LModclmod 20855   LMHom clmhm 21014  LVecclvec 21097

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-lmhm 21017  df-lvec 21098

This theorem is referenced by:  imlmhm  33765  dimkerim  33771  algextdeglem8  33868