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Theorem ismgmALT 44325
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 6651 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2syl6eqr 2877 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6651 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2877 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 5060 . 2 (𝑚 = 𝑀 → ((+g𝑚) clLaw (Base‘𝑚) ↔ clLaw 𝐵))
8 df-mgm2 44321 . 2 MgmALT = {𝑚 ∣ (+g𝑚) clLaw (Base‘𝑚)}
97, 8elab2g 3653 1 (𝑀𝑉 → (𝑀 ∈ MgmALT ↔ clLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115   class class class wbr 5047  cfv 6336  Basecbs 16472  +gcplusg 16554   clLaw ccllaw 44285  MgmALTcmgm2 44317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-br 5048  df-iota 6295  df-fv 6344  df-mgm2 44321
This theorem is referenced by:  mgm2mgm  44329
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