Theorem isnsgrp 13774

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 1031	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑋 ∈ 𝐵)
2		oveq1 6092	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ 𝑦) = (𝑋 ⚬ 𝑦))
3	2	oveq1d 6100	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑦) ⚬ 𝑧))
4		oveq1 6092	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝑥 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)))
5	3, 4	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
6	5	notbid 677	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
7	6	rexbidv 2551	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ∃𝑧 ∈ 𝐵 ¬ ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧))))
8	7	rexbidv 2551	. . . . . . . 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 1032	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑌 ∈ 𝐵)
11		oveq2 6093	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ 𝑦) = (𝑋 ⚬ 𝑌))
12	11	oveq1d 6100	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → ((𝑋 ⚬ 𝑦) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑧))
13		oveq1 6092	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑌 → (𝑦 ⚬ 𝑧) = (𝑌 ⚬ 𝑧))
14	13	oveq2d 6101	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑌 → (𝑋 ⚬ (𝑦 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)))
15	12, 14	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑌 → (((𝑋 ⚬ 𝑦) ⚬ 𝑧) = (𝑋 ⚬ (𝑦 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧))))
16	15	notbid 677	. . . . . . . . . 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 2551	. . . . . . . 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 1033	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → 𝑍 ∈ 𝐵)
20		oveq2 6093	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → ((𝑋 ⚬ 𝑌) ⚬ 𝑧) = ((𝑋 ⚬ 𝑌) ⚬ 𝑍))
21		oveq2 6093	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑍 → (𝑌 ⚬ 𝑧) = (𝑌 ⚬ 𝑍))
22	21	oveq2d 6101	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝑋 ⚬ (𝑌 ⚬ 𝑧)) = (𝑋 ⚬ (𝑌 ⚬ 𝑍)))
23	20, 22	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → (((𝑋 ⚬ 𝑌) ⚬ 𝑧) = (𝑋 ⚬ (𝑌 ⚬ 𝑧)) ↔ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) = (𝑋 ⚬ (𝑌 ⚬ 𝑍))))
24	23	notbid 677	. . . . . . . . . 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 2442	. . . . . . . . . 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 2935	. . . . . . . 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 2935	. . . . . . 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 2935	. . . . . 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 2539	. . . . . . . . 9 ⊢ (∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
32	31	reximi 2647	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑧 ∈ 𝐵 ¬ ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ∃𝑦 ∈ 𝐵 ¬ ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
33		rexnalim 2539	. . . . . . . 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 2647	. . . . . 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 2539	. . . . . 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 943	. . . 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 13771	. . . 4 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
42	38, 41	sylnibr 688	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ((𝑋 ⚬ 𝑌) ⚬ 𝑍) ≠ (𝑋 ⚬ (𝑌 ⚬ 𝑍))) → ¬ 𝑀 ∈ Smgrp)
43		df-nel 2516	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ≠ wne 2420   ∉ wnel 2515  ∀wral 2528  ∃wrex 2529  ‘cfv 5377  (class class class)co 6085  Basecbs 13404  +_gcplusg 13484  Mgmcmgm 13727  Smgrpcsgrp 13769

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8271  ax-resscn 8272  ax-1re 8274  ax-addrcl 8277

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088  df-inn 9308  df-2 9366  df-ndx 13407  df-slot 13408  df-base 13410  df-plusg 13497  df-sgrp 13770

This theorem is used by: (None)