Theorem mgm2mgm 48510

Description: Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)

Assertion

Ref	Expression
mgm2mgm	⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)

Proof of Theorem mgm2mgm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2735	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	ismgmALT 48506	. . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
4		fvex 6846	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
5		fvex 6846	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
6		iscllaw 48472	. . . . . 6 ⊢ (((+g‘𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)))
7	4, 5, 6	mp2an 693	. . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
8	1, 2	ismgm 18568	. . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)))
9	8	biimprd 248	. . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm))
10	7, 9	biimtrid 242	. . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm))
11	3, 10	sylbid 240	. . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm))
12	11	pm2.43i 52	. 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)
13		mgmplusgiopALT 48477	. . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
14	1, 2	ismgmALT 48506	. . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
15	13, 14	mpbird 257	. 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT)
16	12, 15	impbii 209	1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  ∀wral 3050  Vcvv 3439   class class class wbr 5097  ‘cfv 6491  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  Mgmcmgm 18565   clLaw ccllaw 48466  MgmALTcmgm2 48498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rab 3399  df-v 3441  df-sbc 3740  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-iota 6447  df-fv 6499  df-ov 7361  df-mgm 18567  df-cllaw 48469  df-mgm2 48502

This theorem is referenced by:  sgrp2sgrp  48511