Theorem grpmgmd 19003

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 18988	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
3		mndmgm 18781	. 2 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4	2, 3	syl 17	1 ⊢ (𝜑 → 𝐺 ∈ Mgm)

Syntax hints:   → wi 4   ∈ wcel 2108  Mgmcmgm 18678  Mndcmnd 18774  Grpcgrp 18975

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-ext 2711  ax-nul 5324

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-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  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-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6527  df-fv 6583  df-ov 7453  df-sgrp 18759  df-mnd 18775  df-grp 18978

This theorem is referenced by:  psrgrpOLD  22002  psrlmod  22005  psrdi  22010  psrdir  22011  mplsubglem  22044  psdmul  22195  psd1  22196  ofldchr  33311