Theorem grpmgmd 19029

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 19014	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
3		mndmgm 18800	. 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4	2, 3	syl 18	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2143  Mgmcmgm 18697  Mndcmnd 18793  Grpcgrp 19001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415  df-sgrp 18778  df-mnd 18794  df-grp 19004

This theorem is referenced by:  rng1zr  20261  ofldchr  21707  psrlmod  22090  psrdi  22095  psrdir  22096  mplsubglem  22129  psdmul  22310  psd1  22311  psdpw  22314