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Theorem mndid 17219
Description: A monoid has a two-sided identity element. (Contributed by NM, 16-Aug-2011.)
Hypotheses
Ref Expression
mndcl.b 𝐵 = (Base‘𝐺)
mndcl.p + = (+g𝐺)
Assertion
Ref Expression
mndid (𝐺 ∈ Mnd → ∃𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥, + ,𝑢   𝑢,𝐵   𝑢,𝐺   𝑢, +

Proof of Theorem mndid
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mndcl.b . . 3 𝐵 = (Base‘𝐺)
2 mndcl.p . . 3 + = (+g𝐺)
31, 2ismnd 17213 . 2 (𝐺 ∈ Mnd ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)))
43simprbi 480 1 (𝐺 ∈ Mnd → ∃𝑢𝐵𝑥𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  Mndcmnd 17210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754  ax-pow 4808
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5813  df-fv 5858  df-ov 6608  df-mgm 17158  df-sgrp 17200  df-mnd 17211
This theorem is referenced by:  mndideu  17220  mndidcl  17224  mndlrid  17226  prds0g  17240
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