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Theorem issgrpALT 42660
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 6411 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2syl6eqr 2851 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6411 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2851 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 4856 . 2 (𝑚 = 𝑀 → ((+g𝑚) assLaw (Base‘𝑚) ↔ assLaw 𝐵))
8 df-sgrp2 42656 . 2 SGrpALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) assLaw (Base‘𝑚)}
97, 8elrab2 3560 1 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157   class class class wbr 4843  cfv 6101  Basecbs 16184  +gcplusg 16267   assLaw casslaw 42619  MgmALTcmgm2 42650  SGrpALTcsgrp2 42652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-sgrp2 42656
This theorem is referenced by:  sgrp2sgrp  42663
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