Theorem ismgmALT 49047

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 6885	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	eqtr4di 2818	. . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ )
4		fveq2 6885	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2818	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	breq12d 5124	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) clLaw (Base‘𝑚) ↔ ⚬ clLaw 𝐵))
8		df-mgm2 49043	. 2 ⊢ MgmALT = {𝑚 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
9	7, 8	elab2g 3641	1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   class class class wbr 5111  ‘cfv 6540  Basecbs 17293  +_gcplusg 17334   clLaw ccllaw 49007  MgmALTcmgm2 49039

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-mgm2 49043

This theorem is used by:  mgm2mgm  49051