Theorem mgm2mgm 48847

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 2765	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2765	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	ismgmALT 48843	. . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
4		fvex 6884	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
5		fvex 6884	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
6		iscllaw 48809	. . . . . 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 704	. . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
8	1, 2	ismgm 18689	. . . . . 6 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀)))
9	8	biimprd 251	. . . . 5 ⊢ (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm))
10	7, 9	biimtrid 245	. . . 4 ⊢ (𝑀 ∈ MgmALT → ((+g‘𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm))
11	3, 10	sylbid 243	. . 3 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm))
12	11	pm2.43i 53	. 2 ⊢ (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)
13		mgmplusgiopALT 48814	. . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
14	1, 2	ismgmALT 48843	. . 3 ⊢ (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
15	13, 14	mpbird 260	. 2 ⊢ (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT)
16	12, 15	impbii 212	1 ⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)

Syntax hints:   ↔ wb 209   ∈ wcel 2145  ∀wral 3079  Vcvv 3457   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  Mgmcmgm 18686   clLaw ccllaw 48803  MgmALTcmgm2 48835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-iota 6481  df-fv 6533  df-ov 7403  df-mgm 18688  df-cllaw 48806  df-mgm2 48839

This theorem is referenced by:  sgrp2sgrp  48848