Theorem dfgrp3e 19256

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 19255	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
4		simp2 1155	. . . 4 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐵 ≠ ∅)
5		sgrpmgm 18919	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
6	5	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) → 𝐺 ∈ Mgm)
7	6	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝐺 ∈ Mgm)
8		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
9	8	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
10		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
11	1, 2	mgmcl 18825	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
12	7, 9, 10, 11	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
13	12	adantr 486	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → (𝑥 + 𝑦) ∈ 𝐵)
14	1, 2	sgrpass 18920	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
15	14	3anassrs 1381	. . . . . . . . . . . 12 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
16	15	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ Smgrp ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
17	16	adantr 486	. . . . . . . . . 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 490	. . . . . . . . . 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 1146	. . . . . . . . 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 418	. . . . . . . 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 3174	. . . . . . 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 3174	. . . . . 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 26	. . . . 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 1128	. . . 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 521	. . 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 4299	. . . . . 6 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑎 𝑎 ∈ 𝐵)
27		3simpa 1166	. . . . . . . . 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 3132	. . . . . . . 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 18917	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐵 → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
30	28, 29	imbitrrid 249	. . . . . . 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 1962	. . . . . 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 220	. . . . 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 412	. . . 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 488	. . . 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 1156	. . . . . 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 3132	. . . . 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 487	. . . 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 1146	. . 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 212	. 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 278	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  +_gcplusg 17434  Mgmcmgm 18820  Smgrpcsgrp 18913  Grpcgrp 19150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-0g 17618  df-mgm 18822  df-sgrp 18914  df-mnd 18930  df-grp 19153  df-minusg 19154  df-sbg 19155

This theorem is used by: (None)