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Mathbox for Thierry Arnoux |
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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 20876 | . . 3 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod) |
3 | eqid 2731 | . . . . 5 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉) | |
4 | eqid 2731 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
5 | 3, 4 | lmhmsca 20874 | . . . 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 20949 | . . . 4 ⊢ (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing) |
8 | 7 | adantr 480 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing) |
9 | 6, 8 | eqeltrd 2832 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing) |
10 | 4 | islvec 20948 | . 2 ⊢ (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing)) |
11 | 2, 9, 10 | sylanbrc 582 | 1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Scalarcsca 17207 DivRingcdr 20583 LModclmod 20702 LMHom clmhm 20863 LVecclvec 20946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-lmhm 20866 df-lvec 20947 |
This theorem is referenced by: imlmhm 33160 dimkerim 33166 algextdeglem8 33235 |
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