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Theorem iscsgrpALT 43961
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 6666 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 fveq2 6666 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
31, 2breq12d 5075 . . 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 5071 . . 3 ( comLaw 𝐵 ↔ (+g𝑀) comLaw (Base‘𝑀))
73, 6syl6bbr 290 . 2 (𝑚 = 𝑀 → ((+g𝑚) comLaw (Base‘𝑚) ↔ comLaw 𝐵))
8 df-csgrp2 43957 . 2 CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g𝑚) comLaw (Base‘𝑚)}
97, 8elrab2 3686 1 (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ comLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1530  wcel 2107   class class class wbr 5062  cfv 6351  Basecbs 16475  +gcplusg 16557   comLaw ccomlaw 43920  SGrpALTcsgrp2 43952  CSGrpALTccsgrp2 43953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-iota 6311  df-fv 6359  df-csgrp2 43957
This theorem is referenced by: (None)
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