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Theorem issgrpALT 41651
 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 6103 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2syl6eqr 2662 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6103 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2662 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 4596 . 2 (𝑚 = 𝑀 → ((+g𝑚) assLaw (Base‘𝑚) ↔ assLaw 𝐵))
8 df-sgrp2 41647 . 2 SGrpALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) assLaw (Base‘𝑚)}
97, 8elrab2 3333 1 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  +gcplusg 15768   assLaw casslaw 41610  MgmALTcmgm2 41641  SGrpALTcsgrp2 41643 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 4373  df-br 4584  df-iota 5768  df-fv 5812  df-sgrp2 41647 This theorem is referenced by:  sgrp2sgrp  41654
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