Theorem lmhmghmd 33001

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 21029	. 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 2104  (class class class)co 7425   GrpHom cghm 19228   LMHom clmhm 21017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6510  df-fun 6560  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-lmhm 21020

This theorem is referenced by:  r1pquslmic  33574  lvecendof1f1o  33624  algextdeglem8  33693