Theorem isnmnd 18603

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	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
isnmnd	⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)

Distinct variable groups:   𝑥,𝐵,𝑧   𝑥,𝑀,𝑧   𝑥, ⚬ ,𝑧

Proof of Theorem isnmnd

Step	Hyp	Ref	Expression
1		neneq 2945	. . . . . . . 8 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑧 ⚬ 𝑥) = 𝑥)
2	1	intnanrd 490	. . . . . . 7 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
3	2	reximi 3083	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
4	3	ralimi 3082	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
5		rexnal 3099	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
6	5	ralbii 3092	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
7		ralnex 3071	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
8	6, 7	bitri 274	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
9	4, 8	sylib 217	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
10	9	intnand 489	. . 3 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑀 ∈ Smgrp ∧ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥)))
11		isnmnd.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		isnmnd.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
13	11, 12	ismnddef 18601	. . 3 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥)))
14	10, 13	sylnibr 328	. 2 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15		df-nel 3046	. 2 ⊢ (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
16	14, 15	sylibr 233	1 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2939   ∉ wnel 3045  ∀wral 3060  ∃wrex 3069  ‘cfv 6529  (class class class)co 7390  Basecbs 17123  +_gcplusg 17176  Smgrpcsgrp 18588  Mndcmnd 18599

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-nul 5296

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3430  df-v 3472  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4520  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-br 5139  df-iota 6481  df-fv 6537  df-ov 7393  df-mnd 18600

This theorem is referenced by:  sgrp2nmndlem5  18782  copisnmnd  46337  nnsgrpnmnd  46346  2zrngnring  46484