Theorem isnsgrp 13611

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 1027	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑋 ∈ 𝐵)
2		oveq1 6056	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ 𝑦) = (𝑋 ⚬ 𝑦))
3	2	oveq1d 6064	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑦) ⚬ 𝑧))
4		oveq1 6056	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
5	3, 4	eqeq12d 2247	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
6	5	notbid 673	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
7	6	rexbidv 2543	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
8	7	rexbidv 2543	. . . . . . . 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 277	. . . . . . 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 1028	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑌 ∈ 𝐵)
11		oveq2 6057	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ 𝑦) = (𝑋 ⚬ 𝑌))
12	11	oveq1d 6064	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑧))
13		oveq1 6056	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑦 ⚬ 𝑧) = (𝑌 ⚬ 𝑧))
14	13	oveq2d 6065	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
15	12, 14	eqeq12d 2247	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
16	15	notbid 673	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
17	16	adantl 277	. . . . . . . . 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 2543	. . . . . . . 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 1029	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑍 ∈ 𝐵)
20		oveq2 6057	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑍))
21		oveq2 6057	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑍 → (𝑌 ⚬ 𝑧) = (𝑌 ⚬ 𝑍))
22	21	oveq2d 6065	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝑋 ⚬ (𝑌 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
23	20, 22	eqeq12d 2247	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
24	23	notbid 673	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
25	24	adantl 277	. . . . . . . . 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 2434	. . . . . . . . . 10 ⊢ (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
27	26	adantl 277	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
28	19, 25, 27	rspcedvd 2926	. . . . . . . 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 2926	. . . . . . 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 2926	. . . . . 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		rexnalim 2531	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
32	31	reximi 2639	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
33		rexnalim 2531	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ¬ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
34	32, 33	syl 14	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ¬ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
35	34	reximi 2639	. . . . . 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)))
36		rexnalim 2531	. . . . . 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)))
37	30, 35, 36	3syl 17	. . . . 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)))
38	37	intnand 939	. . . 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))))
39		issgrpn0.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
40		issgrpn0.o	. . . . 5 ⊢ ⚬ = (+g‘𝑀)
41	39, 40	issgrp 13608	. . . 4 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
42	38, 41	sylnibr 684	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ 𝑀 ∈ Smgrp)
43		df-nel 2508	. . 3 ⊢ (𝑀 ∉ Smgrp ↔ ¬ 𝑀 ∈ Smgrp)
44	42, 43	sylibr 134	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑀 ∉ Smgrp)
45	44	ex 115	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍)) → 𝑀 ∉ Smgrp))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203   ≠ wne 2412   ∉ wnel 2507  ∀wral 2520  ∃wrex 2521  ‘cfv 5351  (class class class)co 6049  Basecbs 13204  +_gcplusg 13282  Mgmcmgm 13559  Smgrpcsgrp 13606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-ov 6052  df-inn 9237  df-2 9295  df-ndx 13207  df-slot 13208  df-base 13210  df-plusg 13295  df-sgrp 13607

This theorem is referenced by: (None)