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

Theorem iscsgrpALT 41644
Description: The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
ismgmALT.b 𝐵 = (Base‘𝑀)
ismgmALT.o = (+g𝑀)
Assertion
Ref Expression
iscsgrpALT (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ comLaw 𝐵))

Proof of Theorem iscsgrpALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6088 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6088 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 4591 . . 3 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ (+g𝑀) comLaw (Base‘𝑀)))
4 ismgmALT.o . . . 4 = (+g𝑀)
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5breq12i 4587 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6syl6bbr 277 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-csgrp2 41640 . 2 CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3333 1 (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ comLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383   = wceq 1475  wcel 1977   class class class wbr 4578  cfv 5790  Basecbs 15644  +gcplusg 15717   comLaw ccomlaw 41603  SGrpALTcsgrp2 41635  CSGrpALTccsgrp2 41636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-br 4579  df-iota 5754  df-fv 5798  df-csgrp2 41640
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator