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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 2731 | . . 3 ⊢ (Scalar‘𝑆) = (Scalar‘𝑆) | |
2 | eqid 2731 | . . 3 ⊢ (Scalar‘𝑇) = (Scalar‘𝑇) | |
3 | 1, 2 | lmhmlem 20873 | . 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (Scalar‘𝑇) = (Scalar‘𝑆)))) |
4 | 3 | simprld 769 | 1 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇)) |
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 GrpHom cghm 19134 LModclmod 20702 LMHom clmhm 20863 |
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 |
This theorem is referenced by: lmhmf 20878 islmhm2 20882 lmhmco 20887 lmhmplusg 20888 lmhmvsca 20889 lmhmf1o 20890 lmhmima 20891 lmhmpreima 20892 reslmhm 20896 reslmhm2 20897 reslmhm2b 20898 lmhmeql 20899 lmimgim 20909 ip0l 21500 ipdir 21503 islindf5 21705 isnmhm2 24590 nmoleub2lem 24962 nmoleub2lem2 24964 nmhmcn 24968 lmhmghmd 32633 lmhmqusker 32976 dimkerim 33168 kercvrlsm 42291 pwssplit4 42297 mendring 42400 |
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