Theorem ismgmALT 42658

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	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
ismgmALT	⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵))

Proof of Theorem ismgmALT

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6411	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	syl6eqr 2851	. . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ )
4		fveq2 6411	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	syl6eqr 2851	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	breq12d 4856	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) clLaw (Base‘𝑚) ↔ ⚬ clLaw 𝐵))
8		df-mgm2 42654	. 2 ⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
9	7, 8	elab2g 3545	1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157   class class class wbr 4843  ‘cfv 6101  Basecbs 16184  +_gcplusg 16267   clLaw ccllaw 42618  MgmALTcmgm2 42650

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-mgm2 42654

This theorem is referenced by:  mgm2mgm  42662