Theorem 2idlcpblrng 21565

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 1210	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Rng)
2		simpl3 1212	. . . . . . . 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 19390	. . . . . . . 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 783	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴𝐸𝐶)
8	6, 7	ersym 8730	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶𝐸𝐴)
9		rngabl 20377	. . . . . . . 8 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
10	9	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Abel)
11		eqid 2760	. . . . . . . . . . . 12 ⊢ (LIdeal‘𝑅) = (LIdeal‘𝑅)
12		eqid 2760	. . . . . . . . . . . 12 ⊢ (oppr‘𝑅) = (oppr‘𝑅)
13		eqid 2760	. . . . . . . . . . . 12 ⊢ (LIdeal‘(oppr‘𝑅)) = (LIdeal‘(oppr‘𝑅))
14		2idlcpblrng.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (2Ideal‘𝑅)
15	11, 12, 13, 14	2idlelb 21546	. . . . . . . . . . 11 ⊢ (𝑆 ∈ 𝐼 ↔ (𝑆 ∈ (LIdeal‘𝑅) ∧ 𝑆 ∈ (LIdeal‘(oppr‘𝑅))))
16	15	simplbi 502	. . . . . . . . . 10 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘𝑅))
17	16	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘𝑅))
18	17	adantr 486	. . . . . . . 8 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘𝑅))
19	3, 11	lidlss 21490	. . . . . . . 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 2760	. . . . . . . 8 ⊢ (-g‘𝑅) = (-g‘𝑅)
22	3, 21, 4	eqgabl 20048	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐶𝐸𝐴 ↔ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)))
23	10, 20, 22	syl2an2r 698	. . . . . 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 1161	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐴 ∈ 𝑋)
26		simprr 785	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵𝐸𝐷)
27	3, 21, 4	eqgabl 20048	. . . . . . 7 ⊢ ((𝑅 ∈ Abel ∧ 𝑆 ⊆ 𝑋) → (𝐵𝐸𝐷 ↔ (𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋 ∧ (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)))
28	10, 20, 27	syl2an2r 698	. . . . . 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 1160	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐵 ∈ 𝑋)
31		2idlcpblrng.t	. . . . 5 ⊢ · = (.r‘𝑅)
32	3, 31	rngcl 20386	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 · 𝐵) ∈ 𝑋)
33	1, 25, 30, 32	syl3anc 1398	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵) ∈ 𝑋)
34	24	simp1d 1160	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐶 ∈ 𝑋)
35	29	simp2d 1161	. . . 4 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝐷 ∈ 𝑋)
36	3, 31	rngcl 20386	. . . 4 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋) → (𝐶 · 𝐷) ∈ 𝑋)
37	1, 34, 35, 36	syl3anc 1398	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐷) ∈ 𝑋)
38		rnggrp 20380	. . . . . . 7 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
39	38	3ad2ant1 1151	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑅 ∈ Grp)
40	39	adantr 486	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑅 ∈ Grp)
41	3, 31	rngcl 20386	. . . . . 6 ⊢ ((𝑅 ∈ Rng ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐶 · 𝐵) ∈ 𝑋)
42	1, 34, 30, 41	syl3anc 1398	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · 𝐵) ∈ 𝑋)
43	3, 21	grpnnncan2 19247	. . . . 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 1399	. . . 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 20393	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) = ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)))
46		eqid 2760	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
47	46	subg0cl 19344	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubGrp‘𝑅) → (0g‘𝑅) ∈ 𝑆)
48	47	3ad2ant3 1153	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (0g‘𝑅) ∈ 𝑆)
49	48	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (0g‘𝑅) ∈ 𝑆)
50	29	simp3d 1162	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐷(-g‘𝑅)𝐵) ∈ 𝑆)
51	46, 3, 31, 11	rnglidlmcl 21495	. . . . . . 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 1405	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐶 · (𝐷(-g‘𝑅)𝐵)) ∈ 𝑆)
53	45, 52	eqeltrrd 2861	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
54		eqid 2760	. . . . . . . 8 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅))
55	3, 31, 12, 54	opprmul 20570	. . . . . . 7 ⊢ (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) = ((𝐴(-g‘𝑅)𝐶) · 𝐵)
56	3, 31, 21, 1, 25, 34, 30	rngsubdir 20394	. . . . . . 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 2807	. . . . . 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 20575	. . . . . . . . 9 ⊢ (𝑅 ∈ Rng → (oppr‘𝑅) ∈ Rng)
59	58	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → (oppr‘𝑅) ∈ Rng)
60	59	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (oppr‘𝑅) ∈ Rng)
61	15	simprbi 503	. . . . . . . . 9 ⊢ (𝑆 ∈ 𝐼 → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
62	61	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
63	62	adantr 486	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → 𝑆 ∈ (LIdeal‘(oppr‘𝑅)))
64	24	simp3d 1162	. . . . . . 7 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴(-g‘𝑅)𝐶) ∈ 𝑆)
65	12, 46	oppr0 20579	. . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘(oppr‘𝑅))
66	12, 3	opprbas 20573	. . . . . . . 8 ⊢ 𝑋 = (Base‘(oppr‘𝑅))
67	65, 66, 54, 13	rnglidlmcl 21495	. . . . . . 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 1405	. . . . . 6 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐵(.r‘(oppr‘𝑅))(𝐴(-g‘𝑅)𝐶)) ∈ 𝑆)
69	57, 68	eqeltrrd 2861	. . . . 5 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐴 · 𝐵)(-g‘𝑅)(𝐶 · 𝐵)) ∈ 𝑆)
70	21	subgsubcl 19348	. . . . 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 1398	. . . 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 2861	. . 3 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → ((𝐶 · 𝐷)(-g‘𝑅)(𝐴 · 𝐵)) ∈ 𝑆)
73	3, 21, 4	eqgabl 20048	. . . 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 698	. . 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 1361	. 2 ⊢ (((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) ∧ (𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷)) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷))
76	75	ex 418	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ∈ 𝐼 ∧ 𝑆 ∈ (SubGrp‘𝑅)) → ((𝐴𝐸𝐶 ∧ 𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420   Er wer 8714  Basecbs 17387  ._rcmulr 17429  0_gc0g 17610  Grpcgrp 19144  -_gcsg 19146  SubGrpcsubg 19330   ~_QG cqg 19332  Abelcabl 19995  Rngcrng 20374  opp_rcoppr 20566  LIdealclidl 21484  2Idealc2idl 21542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-ip 17446  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-eqg 19335  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-oppr 20567  df-lss 21207  df-sra 21448  df-rgmod 21449  df-lidl 21486  df-2idl 21543

This theorem is used by:  2idlcpbl  21566  qus2idrng  21567  qusmulrng  21578