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Theorem mndlrid 17133
 Description: A monoid's identity element is a two-sided identity. (Contributed by NM, 18-Aug-2011.)
Hypotheses
Ref Expression
mndlrid.b 𝐵 = (Base‘𝐺)
mndlrid.p + = (+g𝐺)
mndlrid.o 0 = (0g𝐺)
Assertion
Ref Expression
mndlrid ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Proof of Theorem mndlrid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndlrid.b . 2 𝐵 = (Base‘𝐺)
2 mndlrid.o . 2 0 = (0g𝐺)
3 mndlrid.p . 2 + = (+g𝐺)
41, 3mndid 17126 . 2 (𝐺 ∈ Mnd → ∃𝑦𝐵𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
51, 2, 3, 4mgmlrid 17089 1 ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 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-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  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-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118 This theorem is referenced by:  mndlid  17134  mndrid  17135  gsumvallem2  17195  gsumsubm  17196  srgidmlem  18343  ringidmlem  18393  frlmgsum  19930
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