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Theorem lmhmlvec2 32704
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 20643 . . 3 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ π‘ˆ ∈ LMod)
21adantl 483 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LMod)
3 eqid 2733 . . . . 5 (Scalarβ€˜π‘‰) = (Scalarβ€˜π‘‰)
4 eqid 2733 . . . . 5 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
53, 4lmhmsca 20641 . . . 4 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
65adantl 483 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
73lvecdrng 20716 . . . 4 (𝑉 ∈ LVec β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
87adantr 482 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
96, 8eqeltrd 2834 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) ∈ DivRing)
104islvec 20715 . 2 (π‘ˆ ∈ LVec ↔ (π‘ˆ ∈ LMod ∧ (Scalarβ€˜π‘ˆ) ∈ DivRing))
112, 9, 10sylanbrc 584 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  Scalarcsca 17200  DivRingcdr 20357  LModclmod 20471   LMHom clmhm 20630  LVecclvec 20713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-lmhm 20633  df-lvec 20714
This theorem is referenced by:  imlmhm  32706  dimkerim  32712
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