Theorem grpmgmd 18900

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 18885	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
3		mndmgm 18675	. 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4	2, 3	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2109  Mgmcmgm 18572  Mndcmnd 18668  Grpcgrp 18872

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-sgrp 18653  df-mnd 18669  df-grp 18875

This theorem is referenced by:  psrgrpOLD  21873  psrlmod  21876  psrdi  21881  psrdir  21882  mplsubglem  21915  psdmul  22060  psd1  22061  psdpw  22064  ofldchr  33299