Theorem mgm2mgm 49144

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 2762	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2762	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	ismgmALT 49140	. . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
4		fvex 6895	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
5		fvex 6895	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
6		iscllaw 49106	. . . . . 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 705	. . . . 5 ⊢ ((+g‘𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g‘𝑀)𝑦) ∈ (Base‘𝑀))
8	1, 2	ismgm 18735	. . . . . 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 49111	. . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
14	1, 2	ismgmALT 49140	. . 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)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  +_gcplusg 17346  Mgmcmgm 18732   clLaw ccllaw 49100  MgmALTcmgm2 49132

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-iota 6493  df-fv 6545  df-ov 7419  df-mgm 18734  df-cllaw 49103  df-mgm2 49136

This theorem is used by:  sgrp2sgrp  49145