Theorem mgm2mgm 49400

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 2760	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
2		eqid 2760	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
3	1, 2	ismgmALT 49396	. . . 4 ⊢ (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g‘𝑀) clLaw (Base‘𝑀)))
4		fvex 6898	. . . . . 6 ⊢ (+g‘𝑀) ∈ V
5		fvex 6898	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
6		iscllaw 49362	. . . . . 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 18818	. . . . . 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 49367	. . 3 ⊢ (𝑀 ∈ Mgm → (+g‘𝑀) clLaw (Base‘𝑀))
14	1, 2	ismgmALT 49396	. . 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 2144  ∀wral 3076  Vcvv 3450   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17388  +_gcplusg 17429  Mgmcmgm 18815   clLaw ccllaw 49356  MgmALTcmgm2 49388

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-iota 6494  df-fv 6546  df-ov 7423  df-mgm 18817  df-cllaw 49359  df-mgm2 49392

This theorem is used by:  sgrp2sgrp  49401