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Theorem ismgmALT 41624
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 6178 . . . 4 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
2 ismgmALT.o . . . 4 = (+g𝑀)
31, 2syl6eqr 2672 . . 3 (𝑚 = 𝑀 → (+g𝑚) = )
4 fveq2 6178 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5 ismgmALT.b . . . 4 𝐵 = (Base‘𝑀)
64, 5syl6eqr 2672 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
73, 6breq12d 4657 . 2 (𝑚 = 𝑀 → ((+g𝑚) clLaw (Base‘𝑚) ↔ clLaw 𝐵))
8 df-mgm2 41620 . 2 MgmALT = {𝑚 ∣ (+g𝑚) clLaw (Base‘𝑚)}
97, 8elab2g 3347 1 (𝑀𝑉 → (𝑀 ∈ MgmALT ↔ clLaw 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1481  wcel 1988   class class class wbr 4644  cfv 5876  Basecbs 15838  +gcplusg 15922   clLaw ccllaw 41584  MgmALTcmgm2 41616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-mgm2 41620
This theorem is referenced by:  mgm2mgm  41628
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