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Theorem lmhmlvec2 33374
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 π‘ˆ)) β†’ π‘ˆ ∈ LVec)
Proof of Theorem lmhmlvec2
StepHypRef Expression
1 lmhmlmod2 20921 . . 3 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ π‘ˆ ∈ LMod)
21adantl 480 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LMod)
3 eqid 2725 . . . . 5 (Scalarβ€˜π‘‰) = (Scalarβ€˜π‘‰)
4 eqid 2725 . . . . 5 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
53, 4lmhmsca 20919 . . . 4 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
65adantl 480 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
73lvecdrng 20994 . . . 4 (𝑉 ∈ LVec β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
87adantr 479 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
96, 8eqeltrd 2825 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) ∈ DivRing)
104islvec 20993 . 2 (π‘ˆ ∈ LVec ↔ (π‘ˆ ∈ LMod ∧ (Scalarβ€˜π‘ˆ) ∈ DivRing))
112, 9, 10sylanbrc 581 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  β€˜cfv 6543  (class class class)co 7416  Scalarcsca 17235  DivRingcdr 20628  LModclmod 20747   LMHom clmhm 20908  LVecclvec 20991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7419  df-oprab 7420  df-mpo 7421  df-lmhm 20911  df-lvec 20992
This theorem is referenced by:  imlmhm  33376  dimkerim  33382  algextdeglem8  33449
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