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Theorem ismgmALT 47363
Description: The predicate "is a magma". (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
ismgmALT (𝑀𝑉 → (𝑀 ∈ MgmALT ↔ clLaw 𝐵))

Proof of Theorem ismgmALT
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2eqtr4di 2786 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6902 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5eqtr4di 2786 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 5165 . 2 (𝑚 = 𝑀 → ((+g𝑚) clLaw (Base‘𝑚) ↔ clLaw 𝐵))
8 df-mgm2 47359 . 2 MgmALT = {𝑚 ∣ (+g𝑚) clLaw (Base‘𝑚)}
97, 8elab2g 3671 1 (𝑀𝑉 → (𝑀 ∈ MgmALT ↔ clLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098   class class class wbr 5152  cfv 6553  Basecbs 17187  +gcplusg 17240   clLaw ccllaw 47323  MgmALTcmgm2 47355
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 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-mgm2 47359
This theorem is referenced by:  mgm2mgm  47367
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