Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgm2mgm Structured version   Visualization version   GIF version

Theorem mgm2mgm 44141
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 2824 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
2 eqid 2824 . . . . 5 (+g𝑀) = (+g𝑀)
31, 2ismgmALT 44137 . . . 4 (𝑀 ∈ MgmALT → (𝑀 ∈ MgmALT ↔ (+g𝑀) clLaw (Base‘𝑀)))
4 fvex 6686 . . . . . 6 (+g𝑀) ∈ V
5 fvex 6686 . . . . . 6 (Base‘𝑀) ∈ V
6 iscllaw 44103 . . . . . 6 (((+g𝑀) ∈ V ∧ (Base‘𝑀) ∈ V) → ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀)))
74, 5, 6mp2an 690 . . . . 5 ((+g𝑀) clLaw (Base‘𝑀) ↔ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)(𝑥(+g𝑀)𝑦) ∈ (Base‘𝑀))
81, 2ismgm 17856 . . . . . 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, 9syl5bi 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 44108 . . 3 (𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))
141, 2ismgmALT 44137 . . 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 2113  wral 3141  Vcvv 3497   class class class wbr 5069  cfv 6358  (class class class)co 7159  Basecbs 16486  +gcplusg 16568  Mgmcmgm 17853   clLaw ccllaw 44097  MgmALTcmgm2 44129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-iota 6317  df-fv 6366  df-ov 7162  df-mgm 17855  df-cllaw 44100  df-mgm2 44133
This theorem is referenced by:  sgrp2sgrp  44142
  Copyright terms: Public domain W3C validator