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Theorem ringmgm 20262
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 20261 . 2 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
2 mndmgm 18767 . 2 (𝑅 ∈ Mnd → 𝑅 ∈ Mgm)
31, 2syl 17 1 (𝑅 ∈ Ring → 𝑅 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Mgmcmgm 18664  Mndcmnd 18760  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-ring 20253
This theorem is referenced by:  psdvsca  22186  gsumply1subr  22251
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