Theorem dfgrp3e 19028

Description: Alternate definition of a group as a set with a closed, associative operation, for which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Exercise 1 of [Herstein] p. 57. (Contributed by NM, 5-Dec-2006.) (Revised by AV, 28-Aug-2021.)

Hypotheses

Ref	Expression
dfgrp3.b	⊢ 𝐵 = (Base‘𝐺)
dfgrp3.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
dfgrp3e	⊢ (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))

Distinct variable groups:   𝐵,𝑙,𝑟,𝑥,𝑦,𝑧   𝐺,𝑙,𝑟,𝑥,𝑦,𝑧   + ,𝑙,𝑟,𝑥,𝑦,𝑧

Proof of Theorem dfgrp3e

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfgrp3.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		dfgrp3.p	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	dfgrp3 19027	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
4		simp2 1137	. . . 4 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐵 ≠ ∅)
5		sgrpmgm 18707	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
6	5	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Mgm)
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Mgm)
8		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
9	8	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
10		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
11	1, 2	mgmcl 18626	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
12	7, 9, 10, 11	syl3anc 1372	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑥 + 𝑦) ∈ 𝐵)
14	1, 2	sgrpass 18708	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
15	14	3anassrs 1360	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
16	15	ralrimiva 3133	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
17	16	adantr 480	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
18		simpr 484	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
19	13, 17, 18	3jca 1128	. . . . . . . . 9 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
20	19	ex 412	. . . . . . . 8 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ((∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))
21	20	ralimdva 3154	. . . . . . 7 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))
22	21	ralimdva 3154	. . . . . 6 ⊢ (𝐺 ∈ Smgrp → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))
23	22	a1d 25	. . . . 5 ⊢ (𝐺 ∈ Smgrp → (𝐵 ≠ ∅ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))))
24	23	3imp 1110	. . . 4 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
25	4, 24	jca 511	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))
26		n0 4333	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐵)
27		3simpa 1148	. . . . . . . . 9 ⊢ (((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
28	27	2ralimi 3110	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
29	1, 2	issgrpn0 18705	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐵 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
30	28, 29	imbitrrid 246	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp))
31	30	exlimiv 1929	. . . . . 6 ⊢ (∃𝑎 𝑎 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp))
32	26, 31	sylbi 217	. . . . 5 ⊢ (𝐵 ≠ ∅ → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp))
33	32	imp 406	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) → 𝐺 ∈ Smgrp)
34		simpl 482	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) → 𝐵 ≠ ∅)
35		simp3 1138	. . . . . 6 ⊢ (((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
36	35	2ralimi 3110	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
37	36	adantl 481	. . . 4 ⊢ ((𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
38	33, 34, 37	3jca 1128	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))) → (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
39	25, 38	impbii 209	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))
40	3, 39	bitri 275	1 ⊢ (𝐺 ∈ Grp ↔ (𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050  ∃wrex 3059  ∅c0 4313  ‘cfv 6541  (class class class)co 7413  Basecbs 17230  +_gcplusg 17274  Mgmcmgm 18621  Smgrpcsgrp 18701  Grpcgrp 18921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7996  df-2nd 7997  df-0g 17458  df-mgm 18623  df-sgrp 18702  df-mnd 18718  df-grp 18924  df-minusg 18925  df-sbg 18926

This theorem is referenced by: (None)