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Theorem mndlrid 18216
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 18207 . 2 (𝐺 ∈ Mnd → ∃𝑦𝐵𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
51, 2, 3, 4mgmlrid 18163 1 ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  cfv 6397  (class class class)co 7231  Basecbs 16784  +gcplusg 16826  0gc0g 16968  Mndcmnd 18197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rmo 3070  df-rab 3071  df-v 3422  df-sbc 3709  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-iota 6355  df-fun 6399  df-fv 6405  df-riota 7188  df-ov 7234  df-0g 16970  df-mgm 18138  df-sgrp 18187  df-mnd 18198
This theorem is referenced by:  mndlid  18217  mndrid  18218  gsumvallem2  18284  gsumsubm  18285  srgidmlem  19559  ringidmlem  19612  frlmgsum  20758
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