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Theorem isnmnd 17907
Description: A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)
Hypotheses
Ref Expression
isnmnd.b 𝐵 = (Base‘𝑀)
isnmnd.o = (+g𝑀)
Assertion
Ref Expression
isnmnd (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
Distinct variable groups:   𝑥,𝐵,𝑧   𝑥,𝑀,𝑧   𝑥, ,𝑧

Proof of Theorem isnmnd
StepHypRef Expression
1 neneq 2993 . . . . . . . 8 ((𝑧 𝑥) ≠ 𝑥 → ¬ (𝑧 𝑥) = 𝑥)
21intnanrd 493 . . . . . . 7 ((𝑧 𝑥) ≠ 𝑥 → ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
32reximi 3206 . . . . . 6 (∃𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
43ralimi 3128 . . . . 5 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
5 rexnal 3201 . . . . . . 7 (∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
65ralbii 3133 . . . . . 6 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
7 ralnex 3199 . . . . . 6 (∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
86, 7bitri 278 . . . . 5 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
94, 8sylib 221 . . . 4 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
109intnand 492 . . 3 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ (𝑀 ∈ Smgrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
11 isnmnd.b . . . 4 𝐵 = (Base‘𝑀)
12 isnmnd.o . . . 4 = (+g𝑀)
1311, 12ismnddef 17905 . . 3 (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
1410, 13sylnibr 332 . 2 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15 df-nel 3092 . 2 (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
1614, 15sylibr 237 1 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  wne 2987  wnel 3091  wral 3106  wrex 3107  cfv 6324  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  Smgrpcsgrp 17892  Mndcmnd 17903
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 2770  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-mnd 17904
This theorem is referenced by:  sgrp2nmndlem5  18086  copisnmnd  44427  nnsgrpnmnd  44436  2zrngnring  44574
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