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Theorem mndlrid 17908
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 17899 . 2 (𝐺 ∈ Mnd → ∃𝑦𝐵𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
51, 2, 3, 4mgmlrid 17855 1 ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16461  +gcplusg 16543  0gc0g 16691  Mndcmnd 17889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7088  df-ov 7133  df-0g 16693  df-mgm 17830  df-sgrp 17879  df-mnd 17890
This theorem is referenced by:  mndlid  17909  mndrid  17910  gsumvallem2  17976  gsumsubm  17977  srgidmlem  19248  ringidmlem  19298  frlmgsum  20891
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