Theorem rnglidlmsgrp 14535

Description: The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption 0 ∈ 𝑈 is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)

Hypotheses

Ref	Expression
rnglidlabl.l	⊢ 𝐿 = (LIdeal‘𝑅)
rnglidlabl.i	⊢ 𝐼 = (𝑅 ↾s 𝑈)
rnglidlabl.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
rnglidlmsgrp	⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)

Proof of Theorem rnglidlmsgrp

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnglidlabl.l	. . 3 ⊢ 𝐿 = (LIdeal‘𝑅)
2		rnglidlabl.i	. . 3 ⊢ 𝐼 = (𝑅 ↾s 𝑈)
3		rnglidlabl.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	1, 2, 3	rnglidlmmgm 14534	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Mgm)
5		eqid 2230	. . . . . . . . . 10 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
6	5	rngmgp 13973	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (mulGrp‘𝑅) ∈ Smgrp)
7	6	3ad2ant1 1044	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝑅) ∈ Smgrp)
8	7	adantr 276	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (mulGrp‘𝑅) ∈ Smgrp)
9	1, 2	lidlssbas 14515	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ 𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))
10	9	sseld 3225	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑎 ∈ (Base‘𝐼) → 𝑎 ∈ (Base‘𝑅)))
11	9	sseld 3225	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑏 ∈ (Base‘𝐼) → 𝑏 ∈ (Base‘𝑅)))
12	9	sseld 3225	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐿 → (𝑐 ∈ (Base‘𝐼) → 𝑐 ∈ (Base‘𝑅)))
13	10, 11, 12	3anim123d 1355	. . . . . . . . . . 11 ⊢ (𝑈 ∈ 𝐿 → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
14	13	3ad2ant2 1045	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼)) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅))))
15	14	imp 124	. . . . . . . . 9 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎 ∈ (Base‘𝑅) ∧ 𝑏 ∈ (Base‘𝑅) ∧ 𝑐 ∈ (Base‘𝑅)))
16	15	simp1d 1035	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑎 ∈ (Base‘𝑅))
17		eqid 2230	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
18	5, 17	mgpbasg 13963	. . . . . . . . . 10 ⊢ (𝑅 ∈ Rng → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
19	18	3ad2ant1 1044	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
20	19	adantr 276	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
21	16, 20	eleqtrd 2309	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑎 ∈ (Base‘(mulGrp‘𝑅)))
22	15	simp2d 1036	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑏 ∈ (Base‘𝑅))
23	22, 20	eleqtrd 2309	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑏 ∈ (Base‘(mulGrp‘𝑅)))
24	15	simp3d 1037	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑐 ∈ (Base‘𝑅))
25	24, 20	eleqtrd 2309	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑐 ∈ (Base‘(mulGrp‘𝑅)))
26		eqid 2230	. . . . . . . 8 ⊢ (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
27		eqid 2230	. . . . . . . 8 ⊢ (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
28	26, 27	sgrpass 13514	. . . . . . 7 ⊢ (((mulGrp‘𝑅) ∈ Smgrp ∧ (𝑎 ∈ (Base‘(mulGrp‘𝑅)) ∧ 𝑏 ∈ (Base‘(mulGrp‘𝑅)) ∧ 𝑐 ∈ (Base‘(mulGrp‘𝑅)))) → ((𝑎(+g‘(mulGrp‘𝑅))𝑏)(+g‘(mulGrp‘𝑅))𝑐) = (𝑎(+g‘(mulGrp‘𝑅))(𝑏(+g‘(mulGrp‘𝑅))𝑐)))
29	8, 21, 23, 25, 28	syl13anc 1275	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(+g‘(mulGrp‘𝑅))𝑏)(+g‘(mulGrp‘𝑅))𝑐) = (𝑎(+g‘(mulGrp‘𝑅))(𝑏(+g‘(mulGrp‘𝑅))𝑐)))
30		eqid 2230	. . . . . . . . . 10 ⊢ (.r‘𝑅) = (.r‘𝑅)
31	5, 30	mgpplusgg 13961	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
32	31	3ad2ant1 1044	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
33	32	adantr 276	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (.r‘𝑅) = (+g‘(mulGrp‘𝑅)))
34	33	oveqd 6040	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)𝑏) = (𝑎(+g‘(mulGrp‘𝑅))𝑏))
35		eqidd 2231	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑐 = 𝑐)
36	33, 34, 35	oveq123d 6044	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐) = ((𝑎(+g‘(mulGrp‘𝑅))𝑏)(+g‘(mulGrp‘𝑅))𝑐))
37		eqidd 2231	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → 𝑎 = 𝑎)
38	33	oveqd 6040	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑏(.r‘𝑅)𝑐) = (𝑏(+g‘(mulGrp‘𝑅))𝑐))
39	33, 37, 38	oveq123d 6044	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐)) = (𝑎(+g‘(mulGrp‘𝑅))(𝑏(+g‘(mulGrp‘𝑅))𝑐)))
40	29, 36, 39	3eqtr4d 2273	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐) = (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐)))
41		simp2 1024	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑈 ∈ 𝐿)
42		simp1 1023	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑅 ∈ Rng)
43	2, 30	ressmulrg 13251	. . . . . . . . . 10 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (.r‘𝑅) = (.r‘𝐼))
44	43	eqcomd 2236	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (.r‘𝐼) = (.r‘𝑅))
45	44	oveqd 6040	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (𝑎(.r‘𝐼)𝑏) = (𝑎(.r‘𝑅)𝑏))
46		eqidd 2231	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → 𝑐 = 𝑐)
47	44, 45, 46	oveq123d 6044	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → ((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐))
48		eqidd 2231	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → 𝑎 = 𝑎)
49	44	oveqd 6040	. . . . . . . . 9 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (𝑏(.r‘𝐼)𝑐) = (𝑏(.r‘𝑅)𝑐))
50	44, 48, 49	oveq123d 6044	. . . . . . . 8 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) = (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐)))
51	47, 50	eqeq12d 2245	. . . . . . 7 ⊢ ((𝑈 ∈ 𝐿 ∧ 𝑅 ∈ Rng) → (((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐) = (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐))))
52	41, 42, 51	syl2anc 411	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐) = (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐))))
53	52	adantr 276	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → (((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(.r‘𝑅)𝑏)(.r‘𝑅)𝑐) = (𝑎(.r‘𝑅)(𝑏(.r‘𝑅)𝑐))))
54	40, 53	mpbird 167	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) ∧ (𝑎 ∈ (Base‘𝐼) ∧ 𝑏 ∈ (Base‘𝐼) ∧ 𝑐 ∈ (Base‘𝐼))) → ((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)))
55	54	ralrimivvva 2614	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)))
56		ressex 13171	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿) → (𝑅 ↾s 𝑈) ∈ V)
57	42, 41, 56	syl2anc 411	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑅 ↾s 𝑈) ∈ V)
58	2, 57	eqeltrid 2317	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝐼 ∈ V)
59		eqid 2230	. . . . . 6 ⊢ (mulGrp‘𝐼) = (mulGrp‘𝐼)
60		eqid 2230	. . . . . 6 ⊢ (Base‘𝐼) = (Base‘𝐼)
61	59, 60	mgpbasg 13963	. . . . 5 ⊢ (𝐼 ∈ V → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
62	58, 61	syl 14	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (Base‘𝐼) = (Base‘(mulGrp‘𝐼)))
63		eqid 2230	. . . . . . . . . 10 ⊢ (.r‘𝐼) = (.r‘𝐼)
64	59, 63	mgpplusgg 13961	. . . . . . . . 9 ⊢ (𝐼 ∈ V → (.r‘𝐼) = (+g‘(mulGrp‘𝐼)))
65	58, 64	syl 14	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (.r‘𝐼) = (+g‘(mulGrp‘𝐼)))
66	65	oveqd 6040	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)𝑏) = (𝑎(+g‘(mulGrp‘𝐼))𝑏))
67		eqidd 2231	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑐 = 𝑐)
68	65, 66, 67	oveq123d 6044	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = ((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐))
69		eqidd 2231	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → 𝑎 = 𝑎)
70	65	oveqd 6040	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑏(.r‘𝐼)𝑐) = (𝑏(+g‘(mulGrp‘𝐼))𝑐))
71	65, 69, 70	oveq123d 6044	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐)))
72	68, 71	eqeq12d 2245	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐))))
73	62, 72	raleqbidv 2745	. . . . 5 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ∀𝑐 ∈ (Base‘(mulGrp‘𝐼))((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐))))
74	62, 73	raleqbidv 2745	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ∀𝑏 ∈ (Base‘(mulGrp‘𝐼))∀𝑐 ∈ (Base‘(mulGrp‘𝐼))((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐))))
75	62, 74	raleqbidv 2745	. . 3 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (∀𝑎 ∈ (Base‘𝐼)∀𝑏 ∈ (Base‘𝐼)∀𝑐 ∈ (Base‘𝐼)((𝑎(.r‘𝐼)𝑏)(.r‘𝐼)𝑐) = (𝑎(.r‘𝐼)(𝑏(.r‘𝐼)𝑐)) ↔ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))∀𝑐 ∈ (Base‘(mulGrp‘𝐼))((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐))))
76	55, 75	mpbid 147	. 2 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))∀𝑐 ∈ (Base‘(mulGrp‘𝐼))((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐)))
77		eqid 2230	. . 3 ⊢ (Base‘(mulGrp‘𝐼)) = (Base‘(mulGrp‘𝐼))
78		eqid 2230	. . 3 ⊢ (+g‘(mulGrp‘𝐼)) = (+g‘(mulGrp‘𝐼))
79	77, 78	issgrp 13509	. 2 ⊢ ((mulGrp‘𝐼) ∈ Smgrp ↔ ((mulGrp‘𝐼) ∈ Mgm ∧ ∀𝑎 ∈ (Base‘(mulGrp‘𝐼))∀𝑏 ∈ (Base‘(mulGrp‘𝐼))∀𝑐 ∈ (Base‘(mulGrp‘𝐼))((𝑎(+g‘(mulGrp‘𝐼))𝑏)(+g‘(mulGrp‘𝐼))𝑐) = (𝑎(+g‘(mulGrp‘𝐼))(𝑏(+g‘(mulGrp‘𝐼))𝑐))))
80	4, 76, 79	sylanbrc 417	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑈 ∈ 𝐿 ∧ 0 ∈ 𝑈) → (mulGrp‘𝐼) ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2201  ∀wral 2509  Vcvv 2801  ‘cfv 5328  (class class class)co 6023  Basecbs 13105   ↾_s cress 13106  +_gcplusg 13183  ._rcmulr 13184  0_gc0g 13362  Mgmcmgm 13460  Smgrpcsgrp 13507  mulGrpcmgp 13957  Rngcrng 13969  LIdealclidl 14505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-i2m1 8142  ax-0lt1 8143  ax-0id 8145  ax-rnegex 8146  ax-pre-ltirr 8149  ax-pre-lttrn 8151  ax-pre-ltadd 8153

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-pnf 8221  df-mnf 8222  df-ltxr 8224  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-iress 13113  df-plusg 13196  df-mulr 13197  df-sca 13199  df-vsca 13200  df-ip 13201  df-0g 13364  df-mgm 13462  df-sgrp 13508  df-mnd 13523  df-grp 13609  df-abl 13897  df-mgp 13958  df-rng 13970  df-lssm 14391  df-sra 14473  df-rgmod 14474  df-lidl 14507

This theorem is referenced by:  rnglidlrng  14536