Theorem lmhmghmd 33027

Description: A module homomorphism is a group homomorphism. (Contributed by Thierry Arnoux, 2-Apr-2025.)

Hypothesis

Ref	Expression
lmhmghmd.1	⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))

Assertion

Ref	Expression
lmhmghmd	⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Proof of Theorem lmhmghmd

Step	Hyp	Ref	Expression
1		lmhmghmd.1	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))
2		lmghm 21022	. 2 ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
3	1, 2	syl 17	1 ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Syntax hints:   → wi 4   ∈ wcel 2108  (class class class)co 7429   GrpHom cghm 19226   LMHom clmhm 21010

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6512  df-fun 6561  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-lmhm 21013

This theorem is referenced by:  r1pquslmic  33618  lvecendof1f1o  33671  algextdeglem8  33746