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Theorem isnmnd 18751
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 2946 . . . . . . . 8 ((𝑧 𝑥) ≠ 𝑥 → ¬ (𝑧 𝑥) = 𝑥)
21intnanrd 489 . . . . . . 7 ((𝑧 𝑥) ≠ 𝑥 → ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
32reximi 3084 . . . . . 6 (∃𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
43ralimi 3083 . . . . 5 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
5 rexnal 3100 . . . . . . 7 (∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
65ralbii 3093 . . . . . 6 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
7 ralnex 3072 . . . . . 6 (∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
86, 7bitri 275 . . . . 5 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
94, 8sylib 218 . . . 4 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
109intnand 488 . . 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 18749 . . 3 (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
1410, 13sylnibr 329 . 2 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15 df-nel 3047 . 2 (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
1614, 15sylibr 234 1 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wnel 3046  wral 3061  wrex 3070  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Smgrpcsgrp 18731  Mndcmnd 18747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mnd 18748
This theorem is referenced by:  sgrp2nmndlem5  18942  copisnmnd  48085  nnsgrpnmnd  48094  2zrngnring  48174
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