Theorem issgrpALT 41651

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 6103	. . . 4 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
2		ismgmALT.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	syl6eqr 2662	. . 3 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = ⚬ )
4		fveq2 6103	. . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
5		ismgmALT.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
6	4, 5	syl6eqr 2662	. . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
7	3, 6	breq12d 4596	. 2 ⊢ (𝑚 = 𝑀 → ((+g‘𝑚) assLaw (Base‘𝑚) ↔ ⚬ assLaw 𝐵))
8		df-sgrp2 41647	. 2 ⊢ SGrpALT = {𝑚 ∈ MgmALT ∣ (+g‘𝑚) assLaw (Base‘𝑚)}
9	7, 8	elrab2 3333	1 ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  +_gcplusg 15768   assLaw casslaw 41610  MgmALTcmgm2 41641  SGrpALTcsgrp2 41643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-sgrp2 41647

This theorem is referenced by:  sgrp2sgrp  41654