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Theorem lmhmlvec2 33222
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 20880 . . 3 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ π‘ˆ ∈ LMod)
21adantl 481 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LMod)
3 eqid 2726 . . . . 5 (Scalarβ€˜π‘‰) = (Scalarβ€˜π‘‰)
4 eqid 2726 . . . . 5 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
53, 4lmhmsca 20878 . . . 4 (𝐹 ∈ (𝑉 LMHom π‘ˆ) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
65adantl 481 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘‰))
73lvecdrng 20953 . . . 4 (𝑉 ∈ LVec β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
87adantr 480 . . 3 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘‰) ∈ DivRing)
96, 8eqeltrd 2827 . 2 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ (Scalarβ€˜π‘ˆ) ∈ DivRing)
104islvec 20952 . 2 (π‘ˆ ∈ LVec ↔ (π‘ˆ ∈ LMod ∧ (Scalarβ€˜π‘ˆ) ∈ DivRing))
112, 9, 10sylanbrc 582 1 ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom π‘ˆ)) β†’ π‘ˆ ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  β€˜cfv 6537  (class class class)co 7405  Scalarcsca 17209  DivRingcdr 20587  LModclmod 20706   LMHom clmhm 20867  LVecclvec 20950
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-lmhm 20870  df-lvec 20951
This theorem is referenced by:  imlmhm  33224  dimkerim  33230  algextdeglem8  33301
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