Theorem 2idlcpblrng 21381

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 1208	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2		simpl3 1210	. . . . . . . 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 19246	. . . . . . . 8 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → 𝐸 Er 𝑋)
6	2, 5	syl 18	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐸 Er 𝑋)
7		simprl 782	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴𝐸𝐶)
8	6, 7	ersym 8707	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9		rngabl 20233	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
10	9	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11		eqid 2769	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
12		eqid 2769	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
13		eqid 2769	. . . . . . . . . . . 12 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
14		2idlcpblrng.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (2Ideal‘𝑅)
15	11, 12, 13, 14	2idlelb 21363	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅))))
16	15	simplbi 501	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
17	16	3ad2ant2 1150	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
18	17	adantr 485	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
19	3, 11	lidlss 21314	. . . . . . . 8 ⊢ (𝑆 ∈ (LIdeal‘𝑅) → 𝑆 ⊆ 𝑋)
20	18, 19	syl 18	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ⊆ 𝑋)
21		eqid 2769	. . . . . . . 8 ⊢ (-g‘𝑅) = (-g‘𝑅)
22	3, 21, 4	eqgabl 19904	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐶𝐸𝐴 ↔ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)))
23	10, 20, 22	syl2an2r 697	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶𝐸𝐴 ↔ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)))
24	8, 23	mpbid 235	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆))
25	24	simp2d 1159	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴 ∈ 𝑋)
26		simprr 784	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
27	3, 21, 4	eqgabl 19904	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐵𝐸𝐷 ↔ (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)))
28	10, 20, 27	syl2an2r 697	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵𝐸𝐷 ↔ (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)))
29	26, 28	mpbid 235	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆))
30	29	simp1d 1158	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵 ∈ 𝑋)
31		2idlcpblrng.t	. . . . 5 ⊢ · = (.r‘𝑅)
32	3, 31	rngcl 20242	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
33	1, 25, 30, 32	syl3anc 1396	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
34	24	simp1d 1158	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶 ∈ 𝑋)
35	29	simp2d 1159	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐷 ∈ 𝑋)
36	3, 31	rngcl 20242	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
37	1, 34, 35, 36	syl3anc 1396	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38		rnggrp 20236	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39	38	3ad2ant1 1149	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
40	39	adantr 485	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
41	3, 31	rngcl 20242	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
42	1, 34, 30, 41	syl3anc 1396	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
43	3, 21	grpnnncan2 19103	. . . . 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 1397	. . . 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 20249	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) = ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)))
46		eqid 2769	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
47	46	subg0cl 19200	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑆)
48	47	3ad2ant3 1151	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (0g‘𝑅) ∈ 𝑆)
49	48	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (0g‘𝑅) ∈ 𝑆)
50	29	simp3d 1160	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)
51	46, 3, 31, 11	rnglidlmcl 21319	. . . . . . 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 1403	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) ∈ 𝑆)
53	45, 52	eqeltrrd 2870	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54		eqid 2769	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
55	3, 31, 12, 54	opprmul 20422	. . . . . . 7 ⊢ (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴(-g‘𝑅)𝐶) · 𝐵)
56	3, 31, 21, 1, 25, 34, 30	rngsubdir 20250	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴(-g‘𝑅)𝐶) · 𝐵) = ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)))
57	55, 56	eqtrid 2816	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)))
58	12	opprrng 20427	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (oppr‘𝑅) ∈ Rng)
59	58	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (oppr‘𝑅) ∈ Rng)
60	59	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (oppr‘𝑅) ∈ Rng)
61	15	simprbi 502	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
62	61	3ad2ant2 1150	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
63	62	adantr 485	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
64	24	simp3d 1160	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)
65	12, 46	oppr0 20431	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘(oppr‘𝑅))
66	12, 3	opprbas 20425	. . . . . . . 8 ⊢ 𝑋 = (Base‘(oppr‘𝑅))
67	65, 66, 54, 13	rnglidlmcl 21319	. . . . . . 7 ⊢ ((((oppr‘𝑅) ∈ Rng ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅)) ∧ (0g‘𝑅) ∈ 𝑆) ∧ (𝐵 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) ∈ 𝑆)
68	60, 63, 49, 30, 64, 67	syl32anc 1403	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) ∈ 𝑆)
69	57, 68	eqeltrrd 2870	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
70	21	subgsubcl 19204	. . . . 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)
71	2, 53, 69, 70	syl3anc 1396	. . . 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)
72	44, 71	eqeltrrd 2870	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
73	3, 21, 4	eqgabl 19904	. . . 4 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → ((𝐴 · 𝐵)𝐸(𝐶 · 𝐷) ↔ ((𝐴 · 𝐵) ∈ 𝑋 ∧ (𝐶 · 𝐷) ∈ 𝑋 ∧ ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)))
74	10, 20, 73	syl2an2r 697	. . 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)))
75	33, 37, 72, 74	mpbir3and 1359	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))
76	75	ex 417	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411   Er wer 8691  Basecbs 17269  ._rcmulr 17311  0_gc0g 17492  Grpcgrp 19000  -_gcsg 19002  SubGrpcsubg 19186   ~_QG cqg 19188  Abelcabl 19851  Rngcrng 20230  opp_rcoppr 20418  LIdealclidl 21308  2Idealc2idl 21359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-ip 17328  df-0g 17494  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-grp 19003  df-minusg 19004  df-sbg 19005  df-subg 19189  df-eqg 19191  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-oppr 20419  df-lss 21031  df-sra 21272  df-rgmod 21273  df-lidl 21310  df-2idl 21360

This theorem is referenced by:  2idlcpbl  21382  qus2idrng  21383  qusmulrng  21393