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Theorem ismnd 17994
 Description: The predicate "is a monoid". This is the definig theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 17999), whose operation is associative (so, a semigroup, see also mndass 18000) and has a two-sided neutral element (see mndid 18001). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)
Hypotheses
Ref Expression
ismnd.b 𝐵 = (Base‘𝐺)
ismnd.p + = (+g𝐺)
Assertion
Ref Expression
ismnd (𝐺 ∈ Mnd ↔ (∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Distinct variable groups:   𝐵,𝑎,𝑏,𝑐   𝐵,𝑒,𝑎   𝐺,𝑎,𝑏,𝑐   + ,𝑎,𝑒   + ,𝑏,𝑐
Allowed substitution hint:   𝐺(𝑒)

Proof of Theorem ismnd
StepHypRef Expression
1 ismnd.b . . 3 𝐵 = (Base‘𝐺)
2 ismnd.p . . 3 + = (+g𝐺)
31, 2ismnddef 17993 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
4 rexn0 4406 . . . 4 (∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → 𝐵 ≠ ∅)
5 fvprc 6655 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
61, 5syl5eq 2805 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
76necon1ai 2978 . . . 4 (𝐵 ≠ ∅ → 𝐺 ∈ V)
81, 2issgrpv 17983 . . . 4 (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
94, 7, 83syl 18 . . 3 (∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → (𝐺 ∈ Smgrp ↔ ∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
109pm5.32ri 579 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)) ↔ (∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
113, 10bitri 278 1 (𝐺 ∈ Mnd ↔ (∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071  Vcvv 3409  ∅c0 4227  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  +gcplusg 16637  Smgrpcsgrp 17980  Mndcmnd 17991 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-mgm 17932  df-sgrp 17981  df-mnd 17992 This theorem is referenced by:  mndid  18001  ismndd  18013  mndpropd  18016  pwmnd  18182  mhmmnd  18302  signswmnd  32068  nn0mnd  44865
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