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Theorem mgm2mgm 42464
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 2765 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2765 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2ismgmALT 42460 . . . 4 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
4 fvex 6388 . . . . . 6 (+g𝑀) ∈ V
5 fvex 6388 . . . . . 6 (Base‘𝑀) ∈ V
6 iscllaw 42426 . . . . . 6 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
74, 5, 6mp2an 683 . . . . 5 ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
81, 2ismgm 17511 . . . . . 6 (𝑀 ∈ MgmALT → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
98biimprd 239 . . . . 5 (𝑀 ∈ MgmALT → (∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀) → 𝑀 ∈ Mgm))
107, 9syl5bi 233 . . . 4 (𝑀 ∈ MgmALT → ((+g𝑀) clLaw (Base‘𝑀) → 𝑀 ∈ Mgm))
113, 10sylbid 231 . . 3 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm))
1211pm2.43i 52 . 2 (𝑀 ∈ MgmALT → 𝑀 ∈ Mgm)
13 mgmplusgiopALT 42431 . . 3 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
141, 2ismgmALT 42460 . . 3 (𝑀 ∈ Mgm → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
1513, 14mpbird 248 . 2 (𝑀 ∈ Mgm → 𝑀 ∈ MgmALT)
1612, 15impbii 200 1 (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2155  wral 3055  Vcvv 3350   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mgmcmgm 17508   clLaw ccllaw 42420  MgmALTcmgm2 42452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-iota 6031  df-fv 6076  df-ov 6845  df-mgm 17510  df-cllaw 42423  df-mgm2 42456
This theorem is referenced by:  sgrp2sgrp  42465
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