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Theorem mgm2mgm 48148
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.
StepHypRef Expression
1 eqid 2736 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2736 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2ismgmALT 48144 . . . 4 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
4 fvex 6918 . . . . . 6 (+g𝑀) ∈ V
5 fvex 6918 . . . . . 6 (Base‘𝑀) ∈ V
6 iscllaw 48110 . . . . . 6 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
74, 5, 6mp2an 692 . . . . 5 ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
81, 2ismgm 18655 . . . . . 6 (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
98biimprd 248 . . . . 5 (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm))
107, 9biimtrid 242 . . . 4 (𝑀 ∈ MgmALT → ((+g𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm))
113, 10sylbid 240 . . 3 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm))
1211pm2.43i 52 . 2 (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)
13 mgmplusgiopALT 48115 . . 3 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
141, 2ismgmALT 48144 . . 3 (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
1513, 14mpbird 257 . 2 (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT)
1612, 15impbii 209 1 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2107  wral 3060  Vcvv 3479   class class class wbr 5142  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Mgmcmgm 18652   clLaw ccllaw 48104  MgmALTcmgm2 48136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-iota 6513  df-fv 6568  df-ov 7435  df-mgm 18654  df-cllaw 48107  df-mgm2 48140
This theorem is referenced by:  sgrp2sgrp  48149
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