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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 19800 | . . 3 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → 𝑈 ∈ LMod) | |
2 | 1 | adantl 484 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LMod) |
3 | eqid 2820 | . . . . 5 ⊢ (Scalar‘𝑉) = (Scalar‘𝑉) | |
4 | eqid 2820 | . . . . 5 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
5 | 3, 4 | lmhmsca 19798 | . . . 4 ⊢ (𝐹 ∈ (𝑉 LMHom 𝑈) → (Scalar‘𝑈) = (Scalar‘𝑉)) |
6 | 5 | adantl 484 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) = (Scalar‘𝑉)) |
7 | 3 | lvecdrng 19873 | . . . 4 ⊢ (𝑉 ∈ LVec → (Scalar‘𝑉) ∈ DivRing) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑉) ∈ DivRing) |
9 | 6, 8 | eqeltrd 2912 | . 2 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → (Scalar‘𝑈) ∈ DivRing) |
10 | 4 | islvec 19872 | . 2 ⊢ (𝑈 ∈ LVec ↔ (𝑈 ∈ LMod ∧ (Scalar‘𝑈) ∈ DivRing)) |
11 | 2, 9, 10 | sylanbrc 585 | 1 ⊢ ((𝑉 ∈ LVec ∧ 𝐹 ∈ (𝑉 LMHom 𝑈)) → 𝑈 ∈ LVec) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6352 (class class class)co 7153 Scalarcsca 16564 DivRingcdr 19498 LModclmod 19630 LMHom clmhm 19787 LVecclvec 19870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3495 df-sbc 3771 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-id 5457 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-iota 6311 df-fun 6354 df-fv 6360 df-ov 7156 df-oprab 7157 df-mpo 7158 df-lmhm 19790 df-lvec 19871 |
This theorem is referenced by: imlmhm 31043 dimkerim 31047 |
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