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Theorem mgm2mgm 48787
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 2752 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2752 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2ismgmALT 48783 . . . 4 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
4 fvex 6865 . . . . . 6 (+g𝑀) ∈ V
5 fvex 6865 . . . . . 6 (Base‘𝑀) ∈ V
6 iscllaw 48749 . . . . . 6 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
74, 5, 6mp2an 700 . . . . 5 ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
81, 2ismgm 18647 . . . . . 6 (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
98biimprd 250 . . . . 5 (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm))
107, 9biimtrid 244 . . . 4 (𝑀 ∈ MgmALT → ((+g𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm))
113, 10sylbid 242 . . 3 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm))
1211pm2.43i 52 . 2 (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)
13 mgmplusgiopALT 48754 . . 3 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
141, 2ismgmALT 48783 . . 3 (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
1513, 14mpbird 259 . 2 (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT)
1612, 15impbii 211 1 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2132  wral 3066  Vcvv 3444   class class class wbr 5090  cfv 6506  (class class class)co 7381  Basecbs 17217  +gcplusg 17258  Mgmcmgm 18644   clLaw ccllaw 48743  MgmALTcmgm2 48775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-ral 3067  df-rab 3405  df-v 3446  df-sbc 3736  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-iota 6462  df-fv 6514  df-ov 7384  df-mgm 18646  df-cllaw 48746  df-mgm2 48779
This theorem is referenced by:  sgrp2sgrp  48788
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