Theorem iscsgrpALT 47728

Description: The predicate "is a commutative semigroup". (Contributed by AV, 20-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
iscsgrpALT	⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵))

Proof of Theorem iscsgrpALT

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6904	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		fveq2 6904	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3	1, 2	breq12d 5167	. . 3 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ (+g‘𝑀) comLaw (Base‘𝑀)))
4		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	breq12i 5163	. . 3 ⊢ ( ⚬ comLaw 𝐵 ↔ (+g‘𝑀) comLaw (Base‘𝑀))
7	3, 6	bitr4di 288	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) comLaw (Base‘𝑚) ↔ ⚬ comLaw 𝐵))
8		df-csgrp2 47724	. 2 ⊢ CSGrpALT = {𝑚 ∈ SGrpALT ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
9	7, 8	elrab2 3695	1 ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2100   class class class wbr 5154  ‘cfv 6557  Basecbs 17227  +_gcplusg 17280   comLaw ccomlaw 47687  SGrpALTcsgrp2 47719  CSGrpALTccsgrp2 47720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-ext 2700

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2062  df-clab 2707  df-cleq 2721  df-clel 2806  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-ss 3975  df-nul 4334  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-iota 6509  df-fv 6565  df-csgrp2 47724

This theorem is referenced by: (None)