Theorem lmhmghmd 33015

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 21047	. 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 7443   GrpHom cghm 19246   LMHom clmhm 21035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-iota 6520  df-fun 6570  df-fv 6576  df-ov 7446  df-oprab 7447  df-mpo 7448  df-lmhm 21038

This theorem is referenced by:  r1pquslmic  33588  lvecendof1f1o  33638  algextdeglem8  33707