![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > lmhmlvec2 | Structured version Visualization version GIF version |
Description: A homomorphism of left vector spaces has a left vector space as codomain. (Contributed by Thierry Arnoux, 7-May-2023.) |
Ref | Expression |
---|---|
lmhmlvec2 | ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmhmlmod2 19797 | . . 3 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod) | |
2 | 1 | adantl 485 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod) |
3 | eqid 2798 | . . . . 5 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉) | |
4 | eqid 2798 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
5 | 3, 4 | lmhmsca 19795 | . . . 4 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑉)) |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) = (Scalar‘𝑉)) |
7 | 3 | lvecdrng 19870 | . . . 4 ⊢ (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing) |
8 | 7 | adantr 484 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing) |
9 | 6, 8 | eqeltrd 2890 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing) |
10 | 4 | islvec 19869 | . 2 ⊢ (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing)) |
11 | 2, 9, 10 | sylanbrc 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 2111 ‘cfv 6324 (class class class)co 7135 Scalarcsca 16560 DivRingcdr 19495 LModclmod 19627 LMHom clmhm 19784 LVecclvec 19867 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
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 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-lmhm 19787 df-lvec 19868 |
This theorem is referenced by: imlmhm 31107 dimkerim 31111 |
Copyright terms: Public domain | W3C validator |