Theorem imasabl 19895

Description: The image structure of an abelian group is an abelian group (imasgrp 19075 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 19805	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
8		imasabl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	1, 2, 3, 4, 5, 7, 8	imasgrp 19075	. . 3 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
10	1, 2, 4, 6	imasbas 17558	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
11	10	eqcomd 2742	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) = 𝐵)
12	11	eleq2d 2826	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥 ∈ 𝐵))
13	11	eleq2d 2826	. . . . . . . . 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 6969	. . . . . . . . . . . 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 6969	. . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . . . . . . 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 19818	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎))
37	23, 28, 33, 36	syl3anc 1372	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎))
38	37	fveq2d 6909	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎(+g‘𝑅)𝑏) = (𝑎 + 𝑏))
45	44	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑎 + 𝑏)))
46	43	oveqd 7449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑝(+g‘𝑅)𝑞) = (𝑝 + 𝑞))
47	46	fveq2d 6909	. . . . . . . . . . . . . . . . . . . . 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 17578	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏)))
53	39, 41, 42, 52	syl3anc 1372	. . . . . . . . . . . . . . . 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 17578	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎)))
55	39, 42, 41, 54	syl3anc 1372	. . . . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . 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 3154	. . . . . . . . . . 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 3154	. . . . . . . . 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 3201	. . . 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 19809	. . . 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 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  –onto→wfo 6558  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485   “_s cimas 17550  Grpcgrp 18952  Abelcabl 19800

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 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-0g 17487  df-imas 17554  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-minusg 18956  df-cmn 19801  df-abl 19802

This theorem is referenced by:  imasrng  20175