MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ismnd Structured version   Visualization version   GIF version

Theorem ismnd 17237
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 17241), whose operation is associative (so, a semigroup, see also mndass 17242) and has a two-sided neutral element (see mndid 17243). (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 17236 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
4 rexn0 4052 . . . 4 (∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → 𝐵 ≠ ∅)
5 fvprc 6152 . . . . . 6 𝐺 ∈ V → (Base‘𝐺) = ∅)
61, 5syl5eq 2667 . . . . 5 𝐺 ∈ V → 𝐵 = ∅)
76necon1ai 2817 . . . 4 (𝐵 ≠ ∅ → 𝐺 ∈ V)
81, 2issgrpv 17226 . . . 4 (𝐺 ∈ V → (𝐺 ∈ SGrp ↔ ∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
94, 7, 83syl 18 . . 3 (∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → (𝐺 ∈ SGrp ↔ ∀𝑎𝐵𝑏𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
109pm5.32ri 669 . 2 ((𝐺 ∈ SGrp ∧ ∃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 264 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  Vcvv 3190  c0 3897  cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  SGrpcsgrp 17223  Mndcmnd 17234
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4759  ax-pow 4813
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-mgm 17182  df-sgrp 17224  df-mnd 17235
This theorem is referenced by:  mndid  17243  ismndd  17253  mndpropd  17256  mhmmnd  17477  signswmnd  30456
  Copyright terms: Public domain W3C validator