Theorem isnsgrp 18121

Description: A condition for a structure not to be a semigroup. (Contributed by AV, 30-Jan-2020.)

Hypotheses

Ref	Expression
issgrpn0.b	⊢ 𝐵 = (Base‘𝑀)
issgrpn0.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
isnsgrp	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → 𝑀 ∉ Smgrp))

Proof of Theorem isnsgrp

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1193	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑋 ∈ 𝐵)
2		oveq1 7198	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ 𝑦) = (𝑋 ⚬ 𝑦))
3	2	oveq1d 7206	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑦) ⚬ 𝑧))
4		oveq1 7198	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
5	3, 4	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
6	5	notbid 321	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
7	6	rexbidv 3206	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
8	7	rexbidv 3206	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
9	8	adantl 485	. . . . . . 7 ⊢ ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) ∧ 𝑥 = 𝑋) → (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
10		simpl2 1194	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑌 ∈ 𝐵)
11		oveq2 7199	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ 𝑦) = (𝑋 ⚬ 𝑌))
12	11	oveq1d 7206	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑧))
13		oveq1 7198	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑦 ⚬ 𝑧) = (𝑌 ⚬ 𝑧))
14	13	oveq2d 7207	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
15	12, 14	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
16	15	notbid 321	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
17	16	adantl 485	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) ∧ 𝑦 = 𝑌) → (¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
18	17	rexbidv 3206	. . . . . . . 8 ⊢ ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) ∧ 𝑦 = 𝑌) → (∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
19		simpl3 1195	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑍 ∈ 𝐵)
20		oveq2 7199	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑍))
21		oveq2 7199	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑍 → (𝑌 ⚬ 𝑧) = (𝑌 ⚬ 𝑍))
22	21	oveq2d 7207	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝑋 ⚬ (𝑌 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
23	20, 22	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
24	23	notbid 321	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
25	24	adantl 485	. . . . . . . . 9 ⊢ ((((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) ∧ 𝑧 = 𝑍) → (¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
26		neneq 2938	. . . . . . . . . 10 ⊢ (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
27	26	adantl 485	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
28	19, 25, 27	rspcedvd 3530	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
29	10, 18, 28	rspcedvd 3530	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
30	1, 9, 29	rspcedvd 3530	. . . . . 6 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
31		rexnal 3150	. . . . . . . 8 ⊢ (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
32	31	2rexbii 3161	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
33		rexnal2 3169	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
34	32, 33	bitr2i 279	. . . . . 6 ⊢ (¬ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
35	30, 34	sylibr 237	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
36	35	intnand 492	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
37		issgrpn0.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
38		issgrpn0.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
39	37, 38	issgrp 18118	. . . 4 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
40	36, 39	sylnibr 332	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ 𝑀 ∈ Smgrp)
41		df-nel 3037	. . 3 ⊢ (𝑀 ∉ Smgrp ↔ ¬ 𝑀 ∈ Smgrp)
42	40, 41	sylibr 237	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑀 ∉ Smgrp)
43	42	ex 416	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → 𝑀 ∉ Smgrp))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112   ≠ wne 2932   ∉ wnel 3036  ∀wral 3051  ∃wrex 3052  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  +_gcplusg 16749  Mgmcmgm 18066  Smgrpcsgrp 18116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-nul 5184

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-iota 6316  df-fv 6366  df-ov 7194  df-sgrp 18117

This theorem is referenced by:  mgm2nsgrplem4  18302  xrsnsgrp  20353