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Theorem mndlrid 18718
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 18709 . 2 (𝐺 ∈ Mnd → ∃𝑦𝐵𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥))
51, 2, 3, 4mgmlrid 18632 1 ((𝐺 ∈ Mnd ∧ 𝑋𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cfv 6551  (class class class)co 7424  Basecbs 17185  +gcplusg 17238  0gc0g 17426  Mndcmnd 18699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-riota 7380  df-ov 7427  df-0g 17428  df-mgm 18605  df-sgrp 18684  df-mnd 18700
This theorem is referenced by:  mndlid  18719  mndrid  18720  gsumvallem2  18791  gsumsubm  18792  srgidmlem  20146  ringidmlem  20209  frlmgsum  21711
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