Theorem imasabl 19807

Description: The image structure of an abelian group is an abelian group (imasgrp 18988 analog). (Contributed by AV, 22-Feb-2025.)

Hypotheses

Ref	Expression
imasabl.u	⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
imasabl.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasabl.p	⊢ (𝜑 → + = (+g‘𝑅))
imasabl.f	⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
imasabl.e	⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
imasabl.r	⊢ (𝜑 → 𝑅 ∈ Abel)
imasabl.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
imasabl	⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Distinct variable groups:   𝐵,𝑎,𝑏,𝑝,𝑞   𝐹,𝑎,𝑏,𝑝,𝑞   𝑅,𝑝,𝑞   𝑈,𝑎,𝑏,𝑝,𝑞   𝑉,𝑎,𝑏,𝑝,𝑞   + ,𝑝,𝑞   0 ,𝑎,𝑏,𝑝,𝑞   𝜑,𝑎,𝑏,𝑝,𝑞

Allowed substitution hints:   + (𝑎,𝑏)   𝑅(𝑎,𝑏)

Proof of Theorem imasabl

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

Step	Hyp	Ref	Expression
1		imasabl.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))
2		imasabl.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasabl.p	. . . 4 ⊢ (𝜑 → + = (+g‘𝑅))
4		imasabl.f	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
5		imasabl.e	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
6		imasabl.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ Abel)
7	6	ablgrpd 19717	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
8		imasabl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	1, 2, 3, 4, 5, 7, 8	imasgrp 18988	. . 3 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
10	1, 2, 4, 6	imasbas 17435	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
11	10	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) = 𝐵)
12	11	eleq2d 2822	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥 ∈ 𝐵))
13	11	eleq2d 2822	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝑈) ↔ 𝑦 ∈ 𝐵))
14	12, 13	anbi12d 632	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
16		foelcdmi 6895	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥)
17	16	ex 412	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–onto→𝐵 → (𝑥 ∈ 𝐵 → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥))
18		foelcdmi 6895	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉–onto→𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)
19	18	ex 412	. . . . . . . . . . 11 ⊢ (𝐹:𝑉–onto→𝐵 → (𝑦 ∈ 𝐵 → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦))
20	17, 19	anim12d 609	. . . . . . . . . 10 ⊢ (𝐹:𝑉–onto→𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)))
21	4, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)))
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)))
23	6	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑅 ∈ Abel)
24	2	eleq2d 2822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅)))
25	24	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅)))
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅)))
27	26	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅))
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅))
29	2	eleq2d 2822	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅)))
30	29	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
31	30	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
32	31	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
33	32	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ (Base‘𝑅))
34		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝑅) = (+g‘𝑅)
36	34, 35	ablcom 19730	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎))
37	23, 28, 33, 36	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎))
38	37	fveq2d 6838	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑏(+g‘𝑅)𝑎)))
39		simplll 774	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝜑)
40		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
41	40	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
42		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
43	3	eqcomd 2742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (+g‘𝑅) = + )
44	43	oveqd 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎(+g‘𝑅)𝑏) = (𝑎 + 𝑏))
45	44	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏)))
46	43	oveqd 7375	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) = (𝑝 + 𝑞))
47	46	fveq2d 6838	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐹‘(𝑝(+g‘𝑅)𝑞)) = (𝐹‘(𝑝 + 𝑞)))
48	45, 47	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
49	48	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
50	5, 49	sylibrd 259	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞))))
51		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘𝑈) = (+g‘𝑈)
52	4, 50, 1, 2, 6, 35, 51	imasaddval 17455	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏)))
53	39, 41, 42, 52	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏)))
54	4, 50, 1, 2, 6, 35, 51	imasaddval 17455	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎)))
55	39, 42, 41, 54	syl3anc 1373	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎)))
56	38, 53, 55	3eqtr4d 2781	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)))
57	56	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)))
58		oveq12 7367	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑎) = 𝑥 ∧ (𝐹‘𝑏) = 𝑦) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦))
59	58	ancoms 458	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦))
60		oveq12 7367	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝑦(+g‘𝑈)𝑥))
61	59, 60	eqeq12d 2752	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → (((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) ↔ (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
62	61	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → (((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) ↔ (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
63	57, 62	mpbid 232	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) ∧ ((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥)) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))
64	63	exp32 420	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑏) = 𝑦 → ((𝐹‘𝑎) = 𝑥 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))))
65	64	rexlimdva 3137	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → ((𝐹‘𝑎) = 𝑥 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))))
66	65	com23 86	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑎) = 𝑥 → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))))
67	66	rexlimdva 3137	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 → (∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦 → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))))
68	67	impd 410	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
69	22, 68	syld 47	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
70	15, 69	sylbid 240	. . . . . 6 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
71	70	imp 406	. . . . 5 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ (𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈))) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))
72	71	ralrimivva 3179	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))
73		simpr 484	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
74	72, 73	jca 511	. . 3 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))))
75	9, 74	mpdan 687	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))))
76		eqid 2736	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
77	76, 51	isabl2 19721	. . . 4 ⊢ (𝑈 ∈ Abel ↔ (𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
78	77	anbi1i 624	. . 3 ⊢ ((𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)) ↔ ((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
79		an21 644	. . 3 ⊢ (((𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)) ∧ (𝐹‘ 0 ) = (0g‘𝑈)) ↔ (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))))
80	78, 79	bitri 275	. 2 ⊢ ((𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)) ↔ (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))))
81	75, 80	sylibr 234	1 ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  0_gc0g 17361   “_s cimas 17427  Grpcgrp 18865  Abelcabl 19712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-sup 9347  df-inf 9348  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-uz 12754  df-fz 13426  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17139  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-0g 17363  df-imas 17431  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868  df-minusg 18869  df-cmn 19713  df-abl 19714

This theorem is referenced by:  imasrng  20114