Theorem ismnd 17914

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 17919), whose operation is associative (so, a semigroup, see also mndass 17920) and has a two-sided neutral element (see mndid 17921). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)

Hypotheses

Ref	Expression
ismnd.b	⊢ 𝐵 = (Base‘𝐺)
ismnd.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ismnd	⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((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

Step	Hyp	Ref	Expression
1		ismnd.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		ismnd.p	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	ismnddef 17913	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
4		rexn0 4454	. . . 4 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → 𝐵 ≠ ∅)
5		fvprc 6663	. . . . . 6 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
6	1, 5	syl5eq 2868	. . . . 5 ⊢ (¬ 𝐺 ∈ V → 𝐵 = ∅)
7	6	necon1ai 3043	. . . 4 ⊢ (𝐵 ≠ ∅ → 𝐺 ∈ V)
8	1, 2	issgrpv 17903	. . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
9	4, 7, 8	3syl 18	. . 3 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → (𝐺 ∈ Smgrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
10	9	pm5.32ri 578	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)) ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
11	3, 10	bitri 277	1 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494  ∅c0 4291  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  Smgrpcsgrp 17900  Mndcmnd 17911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-nul 5210  ax-pow 5266

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-iota 6314  df-fv 6363  df-ov 7159  df-mgm 17852  df-sgrp 17901  df-mnd 17912

This theorem is referenced by:  mndid  17921  ismndd  17933  mndpropd  17936  pwmnd  18102  mhmmnd  18221  signswmnd  31827  nn0mnd  44106