Theorem grpmgmd 18876

Description: A group is a magma, deduction form. (Contributed by SN, 14-Apr-2025.)

Hypothesis

Ref	Expression
grpmgmd.g	⊢ (𝜑 → 𝐺 ∈ Grp)

Assertion

Ref	Expression
grpmgmd	⊢ (𝜑 → 𝐺 ∈ Mgm)

Proof of Theorem grpmgmd

Step	Hyp	Ref	Expression
1		grpmgmd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
2	1	grpmndd 18861	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
3		mndmgm 18651	. 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4	2, 3	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2113  Mgmcmgm 18548  Mndcmnd 18644  Grpcgrp 18848

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-sgrp 18629  df-mnd 18645  df-grp 18851

This theorem is referenced by:  ofldchr  21515  psrlmod  21898  psrdi  21903  psrdir  21904  mplsubglem  21937  psdmul  22082  psd1  22083  psdpw  22086