Theorem issgrpALT 45419

Description: The predicate "is a semigroup". (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
issgrpALT	⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))

Proof of Theorem issgrpALT

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	eqtr4di 2796	. . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ )
4		fveq2 6774	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2796	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	breq12d 5087	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) assLaw (Base‘𝑚) ↔ ⚬ assLaw 𝐵))
8		df-sgrp2 45415	. 2 ⊢ SGrpALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) assLaw (Base‘𝑚)}
9	7, 8	elrab2 3627	1 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  Basecbs 16912  +_gcplusg 16962   assLaw casslaw 45378  MgmALTcmgm2 45409  SGrpALTcsgrp2 45411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-sgrp2 45415

This theorem is referenced by:  sgrp2sgrp  45422