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Theorem ismnd 17909
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 17914), whose operation is associative (so, a semigroup, see also mndass 17915) and has a two-sided neutral element (see mndid 17916). (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 17908 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
4 rexn0 4457 . . . 4 (∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → 𝐵 ≠ ∅)
5 fvprc 6662 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
61, 5syl5eq 2873 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
76necon1ai 3048 . . . 4 (𝐵 ≠ ∅ → 𝐺 ∈ V)
81, 2issgrpv 17898 . . . 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 576 . 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 276 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 207  wa 396   = wceq 1530  wcel 2107  wne 3021  wral 3143  wrex 3144  Vcvv 3500  c0 4295  cfv 6354  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  Smgrpcsgrp 17895  Mndcmnd 17906
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-nul 5207  ax-pow 5263
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-iota 6313  df-fv 6362  df-ov 7153  df-mgm 17847  df-sgrp 17896  df-mnd 17907
This theorem is referenced by:  mndid  17916  ismndd  17928  mndpropd  17931  pwmnd  18047  mhmmnd  18166  signswmnd  31732  nn0mnd  43937
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