Theorem issgrpALT 47638

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 6893	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	eqtr4di 2784	. . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ )
4		fveq2 6893	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	eqtr4di 2784	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	breq12d 5158	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) assLaw (Base‘𝑚) ↔ ⚬ assLaw 𝐵))
8		df-sgrp2 47634	. 2 ⊢ SGrpALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) assLaw (Base‘𝑚)}
9	7, 8	elrab2 3683	1 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   class class class wbr 5145  ‘cfv 6546  Basecbs 17208  +_gcplusg 17261   assLaw casslaw 47597  MgmALTcmgm2 47628  SGrpALTcsgrp2 47630

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 2101  ax-9 2109  ax-ext 2697

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 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-iota 6498  df-fv 6554  df-sgrp2 47634

This theorem is referenced by:  sgrp2sgrp  47641