Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  smgrpismgmOLD Structured version   Visualization version   GIF version

Theorem smgrpismgmOLD 34287
 Description: Obsolete version of sgrpmgm 17675 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
smgrpismgmOLD (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Proof of Theorem smgrpismgmOLD
StepHypRef Expression
1 elin 4019 . . 3 (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
21simplbi 493 . 2 (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3 df-sgrOLD 34286 . 2 SemiGrp = (Magma ∩ Ass)
42, 3eleq2s 2877 1 (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107   ∩ cin 3791  Asscass 34267  Magmacmagm 34273  SemiGrpcsem 34285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-in 3799  df-sgrOLD 34286 This theorem is referenced by:  mndoismgmOLD  34295
 Copyright terms: Public domain W3C validator