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

Theorem issgrpALT 44852
 Description: The predicate "is a semigroup". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b 𝐵 = (Base‘𝑀)
ismgmALT.o = (+g𝑀)
Assertion
Ref Expression
issgrpALT (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))

Proof of Theorem issgrpALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6658 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2eqtr4di 2811 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6658 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5eqtr4di 2811 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 5045 . 2 (𝑚 = 𝑀 → ((+g𝑚) assLaw (Base‘𝑚) ↔ assLaw 𝐵))
8 df-sgrp2 44848 . 2 SGrpALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) assLaw (Base‘𝑚)}
97, 8elrab2 3605 1 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ‘cfv 6335  Basecbs 16541  +gcplusg 16623   assLaw casslaw 44811  MgmALTcmgm2 44842  SGrpALTcsgrp2 44844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-rab 3079  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-iota 6294  df-fv 6343  df-sgrp2 44848 This theorem is referenced by:  sgrp2sgrp  44855
 Copyright terms: Public domain W3C validator