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Theorem iscsgrpALT 44984
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 6677 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6677 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 5044 . . 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 5040 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6bitr4di 292 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-csgrp2 44980 . 2 CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3592 1 (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ comLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114   class class class wbr 5031  cfv 6340  Basecbs 16589  +gcplusg 16671   comLaw ccomlaw 44943  SGrpALTcsgrp2 44975  CSGrpALTccsgrp2 44976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-rab 3063  df-v 3401  df-un 3849  df-in 3851  df-ss 3861  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-iota 6298  df-fv 6348  df-csgrp2 44980
This theorem is referenced by: (None)
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