MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ringmgm Structured version   Visualization version   GIF version

Theorem ringmgm 18764
Description: A ring is a magma. (Contributed by AV, 31-Jan-2020.)
Assertion
Ref Expression
ringmgm (𝑅 ∈ Ring → 𝑅 ∈ Mgm)

Proof of Theorem ringmgm
StepHypRef Expression
1 ringmnd 18763 . 2 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2 mndmgm 17507 . 2 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
31, 2syl 17 1 (𝑅 ∈ Ring → 𝑅 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2144  Mgmcmgm 17447  Mndcmnd 17501  Ringcrg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795  df-sgrp 17491  df-mnd 17502  df-grp 17632  df-ring 18756
This theorem is referenced by:  gsumply1subr  19818
  Copyright terms: Public domain W3C validator