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Theorem issgrpALT 47175
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 6885 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2eqtr4di 2784 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6885 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5eqtr4di 2784 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 5154 . 2 (𝑚 = 𝑀 → ((+g𝑚) assLaw (Base‘𝑚) ↔ assLaw 𝐵))
8 df-sgrp2 47171 . 2 SGrpALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) assLaw (Base‘𝑚)}
97, 8elrab2 3681 1 (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1533  wcel 2098   class class class wbr 5141  cfv 6537  Basecbs 17153  +gcplusg 17206   assLaw casslaw 47134  MgmALTcmgm2 47165  SGrpALTcsgrp2 47167
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545  df-sgrp2 47171
This theorem is referenced by:  sgrp2sgrp  47178
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