Theorem 2idlcpblrng 14843

Description: The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)

Hypotheses

Ref	Expression
2idlcpblrng.x	⊢ 𝑋 = (Base‘𝑅)
2idlcpblrng.r	⊢ 𝐸 = (𝑅 ~QG 𝑆)
2idlcpblrng.i	⊢ 𝐼 = (2Ideal‘𝑅)
2idlcpblrng.t	⊢ · = (.r‘𝑅)

Assertion

Ref	Expression
2idlcpblrng	⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Proof of Theorem 2idlcpblrng

Step	Hyp	Ref	Expression
1		simpl1 1031	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2		simpl3 1033	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (SubGrp‘𝑅))
3		2idlcpblrng.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑅)
4		2idlcpblrng.r	. . . . . . . . 9 ⊢ 𝐸 = (𝑅 ~QG 𝑆)
5	3, 4	eqger 14010	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → 𝐸 Er 𝑋)
6	2, 5	syl 14	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐸 Er 𝑋)
7		simprl 535	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴𝐸𝐶)
8	6, 7	ersym 6813	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9		rngabl 14217	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
10	9	3ad2ant1 1049	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11		eqid 2238	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
12		eqid 2238	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
13		eqid 2238	. . . . . . . . . . . 12 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
14		2idlcpblrng.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (2Ideal‘𝑅)
15	11, 12, 13, 14	2idlelb 14825	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅))))
16	15	simplbi 274	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
17	16	3ad2ant2 1050	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
18	17	adantr 276	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
19	3, 11	lidlss 14796	. . . . . . . 8 ⊢ (𝑆 ∈ (LIdeal‘𝑅) → 𝑆 ⊆ 𝑋)
20	18, 19	syl 14	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ⊆ 𝑋)
21		eqid 2238	. . . . . . . 8 ⊢ (-g‘𝑅) = (-g‘𝑅)
22	3, 21, 4	eqgabl 14117	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐶𝐸𝐴 ↔ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)))
23	10, 20, 22	syl2an2r 603	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶𝐸𝐴 ↔ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)))
24	8, 23	mpbid 147	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆))
25	24	simp2d 1041	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴 ∈ 𝑋)
26		simprr 537	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
27	3, 21, 4	eqgabl 14117	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐵𝐸𝐷 ↔ (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)))
28	10, 20, 27	syl2an2r 603	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵𝐸𝐷 ↔ (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)))
29	26, 28	mpbid 147	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆))
30	29	simp1d 1040	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵 ∈ 𝑋)
31		2idlcpblrng.t	. . . . 5 ⊢ · = (.r‘𝑅)
32	3, 31	rngcl 14226	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
33	1, 25, 30, 32	syl3anc 1278	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
34	24	simp1d 1040	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶 ∈ 𝑋)
35	29	simp2d 1041	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐷 ∈ 𝑋)
36	3, 31	rngcl 14226	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
37	1, 34, 35, 36	syl3anc 1278	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38		rnggrp 14220	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39	38	3ad2ant1 1049	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
40	39	adantr 276	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
41	3, 31	rngcl 14226	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
42	1, 34, 30, 41	syl3anc 1278	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
43	3, 21	grpnnncan2 13885	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝐶 · 𝐷) ∈ 𝑋 ∧ (𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐵) ∈ 𝑋)) → (((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵))(-g‘𝑅)((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵))) = ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)))
44	40, 37, 33, 42, 43	syl13anc 1280	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵))(-g‘𝑅)((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵))) = ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)))
45	3, 31, 21, 1, 34, 35, 30	rngsubdi 14233	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) = ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)))
46		eqid 2238	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
47	46	subg0cl 13968	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑆)
48	47	3ad2ant3 1051	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (0g‘𝑅) ∈ 𝑆)
49	48	adantr 276	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (0g‘𝑅) ∈ 𝑆)
50	29	simp3d 1042	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)
51	46, 3, 31, 11	rnglidlmcl 14800	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ (LIdeal‘𝑅) ∧ (0g‘𝑅) ∈ 𝑆) ∧ (𝐶 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) ∈ 𝑆)
52	1, 18, 49, 34, 50, 51	syl32anc 1286	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) ∈ 𝑆)
53	45, 52	eqeltrrd 2316	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54	3, 21	grpsubcl 13868	. . . . . . . . 9 ⊢ ((𝑅 ∈ Grp ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴(-g‘𝑅)𝐶) ∈ 𝑋)
55	40, 25, 34, 54	syl3anc 1278	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴(-g‘𝑅)𝐶) ∈ 𝑋)
56		eqid 2238	. . . . . . . . 9 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
57	3, 31, 12, 56	opprmulg 14359	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝐵 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑋) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴(-g‘𝑅)𝐶) · 𝐵))
58	1, 30, 55, 57	syl3anc 1278	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴(-g‘𝑅)𝐶) · 𝐵))
59	3, 31, 21, 1, 25, 34, 30	rngsubdir 14234	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴(-g‘𝑅)𝐶) · 𝐵) = ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)))
60	58, 59	eqtrd 2271	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)))
61	12	opprrng 14365	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (oppr‘𝑅) ∈ Rng)
62	61	3ad2ant1 1049	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (oppr‘𝑅) ∈ Rng)
63	62	adantr 276	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (oppr‘𝑅) ∈ Rng)
64	15	simprbi 275	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
65	64	3ad2ant2 1050	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
66	65	adantr 276	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
67	12, 46	oppr0g 14370	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (0g‘𝑅) = (0g‘(oppr‘𝑅)))
68	1, 67	syl 14	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (0g‘𝑅) = (0g‘(oppr‘𝑅)))
69	68, 49	eqeltrrd 2316	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (0g‘(oppr‘𝑅)) ∈ 𝑆)
70	12, 3	opprbasg 14363	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → 𝑋 = (Base‘(oppr‘𝑅)))
71	1, 70	syl 14	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑋 = (Base‘(oppr‘𝑅)))
72	30, 71	eleqtrd 2317	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵 ∈ (Base‘(oppr‘𝑅)))
73	24	simp3d 1042	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)
74		eqid 2238	. . . . . . . 8 ⊢ (0g‘(oppr‘𝑅)) = (0g‘(oppr‘𝑅))
75		eqid 2238	. . . . . . . 8 ⊢ (Base‘(oppr‘𝑅)) = (Base‘(oppr‘𝑅))
76	74, 75, 56, 13	rnglidlmcl 14800	. . . . . . 7 ⊢ ((((oppr‘𝑅) ∈ Rng ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘(oppr‘𝑅)) ∈ 𝑆) ∧ (𝐵 ∈ (Base‘(oppr‘𝑅)) ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) ∈ 𝑆)
77	63, 66, 69, 72, 73, 76	syl32anc 1286	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) ∈ 𝑆)
78	60, 77	eqeltrrd 2316	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
79	21	subgsubcl 13971	. . . . 5 ⊢ ((𝑆 ∈ (SubGrp‘𝑅) ∧ ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆 ∧ ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆) → (((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵))(-g‘𝑅)((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
80	2, 53, 78, 79	syl3anc 1278	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵))(-g‘𝑅)((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵))) ∈ 𝑆)
81	44, 80	eqeltrrd 2316	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
82	3, 21, 4	eqgabl 14117	. . . 4 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
83	10, 20, 82	syl2an2r 603	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
84	33, 37, 81, 83	mpbir3and 1211	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))
85	84	ex 115	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220   class class class wbr 4128  ‘cfv 5375  (class class class)co 6079   Er wer 6798  Basecbs 13335  ._rcmulr 13415  0_gc0g 13593  Grpcgrp 13788  -_gcsg 13790  SubGrpcsubg 13953   ~_QG cqg 13955  Abelcabl 14071  Rngcrng 14214  opp_rcoppr 14355  LIdealclidl 14787  2Idealc2idl 14819

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-tpos 6510  df-er 6801  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-ip 13432  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-sbg 13793  df-subg 13956  df-eqg 13958  df-cmn 14072  df-abl 14073  df-mgp 14201  df-rng 14215  df-oppr 14356  df-lssm 14673  df-sra 14755  df-rgmod 14756  df-lidl 14789  df-2idl 14820

This theorem is referenced by:  2idlcpbl  14844  qus2idrng  14845  qusmulrng  14852