Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > lmghm | Structured version Visualization version GIF version |
Description: A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.) |
Ref | Expression |
---|---|
lmghm | ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
2 | eqid 2610 | . . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
3 | 1, 2 | lmhmlem 18850 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆)))) |
4 | simprl 790 | . 2 ⊢ (((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆))) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Scalarcsca 15771 GrpHom cghm 17480 LModclmod 18686 LMHom clmhm 18840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-lmhm 18843 |
This theorem is referenced by: lmhmf 18855 islmhm2 18859 lmhmco 18864 lmhmplusg 18865 lmhmvsca 18866 lmhmf1o 18867 lmhmima 18868 lmhmpreima 18869 reslmhm 18873 reslmhm2 18874 reslmhm2b 18875 lmhmeql 18876 lmimgim 18886 ip0l 19800 ipdir 19803 islindf5 19997 isnmhm2 22366 nmoleub2lem 22722 nmoleub2lem2 22724 nmhmcn 22728 kercvrlsm 36671 pwssplit4 36677 mendring 36781 |
Copyright terms: Public domain | W3C validator |