Theorem mgm2mgm 48703

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 2736	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2736	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	ismgmALT 48699	. . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
4		fvex 6853	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
5		fvex 6853	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
6		iscllaw 48665	. . . . . 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 18609	. . . . . 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 48670	. . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
14	1, 2	ismgmALT 48699	. . 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 3051  Vcvv 3429   class class class wbr 5085  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Mgmcmgm 18606   clLaw ccllaw 48659  MgmALTcmgm2 48691

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-iota 6454  df-fv 6506  df-ov 7370  df-mgm 18608  df-cllaw 48662  df-mgm2 48695

This theorem is referenced by:  sgrp2sgrp  48704