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Theorem isnmnd 17497
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 2936 . . . . . . . 8 ((𝑧 𝑥) ≠ 𝑥 → ¬ (𝑧 𝑥) = 𝑥)
21intnanrd 1001 . . . . . . 7 ((𝑧 𝑥) ≠ 𝑥 → ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
32reximi 3147 . . . . . 6 (∃𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
43ralimi 3088 . . . . 5 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
5 rexnal 3131 . . . . . . 7 (∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
65ralbii 3116 . . . . . 6 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
7 ralnex 3128 . . . . . 6 (∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
86, 7bitri 264 . . . . 5 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
94, 8sylib 208 . . . 4 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
109intnand 1000 . . 3 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ (𝑀 ∈ SGrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
11 isnmnd.b . . . 4 𝐵 = (Base‘𝑀)
12 isnmnd.o . . . 4 = (+g𝑀)
1311, 12ismnddef 17495 . . 3 (𝑀 ∈ Mnd ↔ (𝑀 ∈ SGrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
1410, 13sylnibr 318 . 2 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15 df-nel 3034 . 2 (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
1614, 15sylibr 224 1 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1630  wcel 2137  wne 2930  wnel 3033  wral 3048  wrex 3049  cfv 6047  (class class class)co 6811  Basecbs 16057  +gcplusg 16141  SGrpcsgrp 17482  Mndcmnd 17493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-nul 4939
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-nel 3034  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-iota 6010  df-fv 6055  df-ov 6814  df-mnd 17494
This theorem is referenced by:  sgrp2nmndlem5  17615  copisnmnd  42317  nnsgrpnmnd  42326  2zrngnring  42460
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