Theorem imasabl 20117

Description: The image structure of an abelian group is an abelian group (imasgrp 19292 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⟶onto𝐵)
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⟶onto𝐵)
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 20027	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp)
8		imasabl.z	. . . 4 ⊢ 0 = (0g‘𝑅)
9	1, 2, 3, 4, 5, 7, 8	imasgrp 19292	. . 3 ⊢ (𝜑 → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
10	1, 2, 4, 6	imasbas 17710	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (Base‘𝑈))
11	10	eqcomd 2769	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) = 𝐵)
12	11	eleq2d 2850	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑈) ↔ 𝑥 ∈ 𝐵))
13	11	eleq2d 2850	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ (Base‘𝑈) ↔ 𝑦 ∈ 𝐵))
14	12, 13	anbi12d 644	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ (Base‘𝑈) ∧ 𝑦 ∈ (Base‘𝑈)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
16		foelcdmi 6966	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉⟶onto𝐵 ∧ 𝑥 ∈ 𝐵) → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥)
17	16	ex 418	. . . . . . . . . . 11 ⊢ (𝐹:𝑉⟶onto𝐵 → (𝑥 ∈ 𝐵 → ∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥))
18		foelcdmi 6966	. . . . . . . . . . . 12 ⊢ ((𝐹:𝑉⟶onto𝐵 ∧ 𝑦 ∈ 𝐵) → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)
19	18	ex 418	. . . . . . . . . . 11 ⊢ (𝐹:𝑉⟶onto𝐵 → (𝑦 ∈ 𝐵 → ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦))
20	17, 19	anim12d 621	. . . . . . . . . 10 ⊢ (𝐹:𝑉⟶onto𝐵 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)))
21	4, 20	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (∃𝑎 ∈ 𝑉 (𝐹‘𝑎) = 𝑥 ∧ ∃𝑏 ∈ 𝑉 (𝐹‘𝑏) = 𝑦)))
22	21	adantr 486	. . . . . . . 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 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑅 ∈ Abel)
24	2	eleq2d 2850	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑎 ∈ 𝑉 ↔ 𝑎 ∈ (Base‘𝑅)))
25	24	biimpd 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅)))
26	25	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑎 ∈ 𝑉 → 𝑎 ∈ (Base‘𝑅)))
27	26	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅))
28	27	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ (Base‘𝑅))
29	2	eleq2d 2850	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑏 ∈ 𝑉 ↔ 𝑏 ∈ (Base‘𝑅)))
30	29	biimpd 232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
31	30	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
32	31	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → (𝑏 ∈ 𝑉 → 𝑏 ∈ (Base‘𝑅)))
33	32	imp 412	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ (Base‘𝑅))
34		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘𝑅) = (Base‘𝑅)
35		eqid 2763	. . . . . . . . . . . . . . . . . . 19 ⊢ (+g‘𝑅) = (+g‘𝑅)
36	34, 35	ablcom 20040	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ Abel ∧ 𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅)) → (𝑎(+g‘𝑅)𝑏) = (𝑏(+g‘𝑅)𝑎))
37	23, 28, 33, 36	syl3anc 1399	. . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝜑)
40		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) → 𝑎 ∈ 𝑉)
41	40	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑎 ∈ 𝑉)
42		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) ∧ 𝑎 ∈ 𝑉) ∧ 𝑏 ∈ 𝑉) → 𝑏 ∈ 𝑉)
43	3	eqcomd 2769	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (+g‘𝑅) = + )
44	43	oveqd 7457	. . . . . . . . . . . . . . . . . . . . . 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 7457	. . . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ((𝐹‘(𝑎(+g‘𝑅)𝑏)) = (𝐹‘(𝑝(+g‘𝑅)𝑞)) ↔ (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑝 + 𝑞))))
49	48	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . 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 262	. . . . . . . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘𝑈) = (+g‘𝑈)
52	4, 50, 1, 2, 6, 35, 51	imasaddval 17730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝐹‘(𝑎(+g‘𝑅)𝑏)))
53	39, 41, 42, 52	syl3anc 1399	. . . . . . . . . . . . . . . 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 17730	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝐹‘(𝑏(+g‘𝑅)𝑎)))
55	39, 42, 41, 54	syl3anc 1399	. . . . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑎) = 𝑥 ∧ (𝐹‘𝑏) = 𝑦) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦))
59	58	ancoms 464	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑎)(+g‘𝑈)(𝐹‘𝑏)) = (𝑥(+g‘𝑈)𝑦))
60		oveq12 7449	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑏) = 𝑦 ∧ (𝐹‘𝑎) = 𝑥) → ((𝐹‘𝑏)(+g‘𝑈)(𝐹‘𝑎)) = (𝑦(+g‘𝑈)𝑥))
61	59, 60	eqeq12d 2779	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . 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 235	. . . . . . . . . . . . 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 426	. . . . . . . . . . . 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 3167	. . . . . . . . . . 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 87	. . . . . . . . . 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 3167	. . . . . . . . 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 416	. . . . . . . 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 48	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
70	15, 69	sylbid 243	. . . . . 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 412	. . . . 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 3209	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥))
73		simpr 490	. . . 4 ⊢ ((𝜑 ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))) → (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈)))
74	72, 73	jca 521	. . 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 700	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥) ∧ (𝑈 ∈ Grp ∧ (𝐹‘ 0 ) = (0g‘𝑈))))
76		eqid 2763	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
77	76, 51	isabl2 20031	. . . 4 ⊢ (𝑈 ∈ Abel ↔ (𝑈 ∈ Grp ∧ ∀𝑥 ∈ (Base‘𝑈)∀𝑦 ∈ (Base‘𝑈)(𝑥(+g‘𝑈)𝑦) = (𝑦(+g‘𝑈)𝑥)))
78	77	anbi1i 636	. . 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 657	. . 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 278	. 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 237	1 ⊢ (𝜑 → (𝑈 ∈ Abel ∧ (𝐹‘ 0 ) = (0g‘𝑈)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⟶ontowfo 6553  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  0_gc0g 17636   “_s cimas 17702  Grpcgrp 19170  Abelcabl 20022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-sup 9458  df-inf 9459  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-slot 17386  df-ndx 17398  df-base 17414  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-ip 17472  df-tset 17473  df-ple 17474  df-ds 17476  df-0g 17638  df-imas 17706  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-grp 19173  df-minusg 19174  df-cmn 20023  df-abl 20024

This theorem is used by:  imasrng  20426