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Theorem mgm2mgm 48797
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 2756 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2756 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2ismgmALT 48793 . . . 4 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
4 fvex 6869 . . . . . 6 (+g𝑀) ∈ V
5 fvex 6869 . . . . . 6 (Base‘𝑀) ∈ V
6 iscllaw 48759 . . . . . 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 18651 . . . . . 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 48764 . . 3 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
141, 2ismgmALT 48793 . . 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 2136  wral 3070  Vcvv 3448   class class class wbr 5094  cfv 6510  (class class class)co 7385  Basecbs 17221  +gcplusg 17262  Mgmcmgm 18648   clLaw ccllaw 48753  MgmALTcmgm2 48785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pr 5384
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-ral 3071  df-rab 3409  df-v 3450  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-iota 6466  df-fv 6518  df-ov 7388  df-mgm 18650  df-cllaw 48756  df-mgm2 48789
This theorem is referenced by:  sgrp2sgrp  48798
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