Theorem erler 33055

Description: The relation used to build the ring localization is an equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
erler.1	⊢ 𝐵 = (Base‘𝑅)
erler.2	⊢ 0 = (0g‘𝑅)
erler.3	⊢ 1 = (1r‘𝑅)
erler.4	⊢ · = (.r‘𝑅)
erler.5	⊢ − = (-g‘𝑅)
erler.w	⊢ 𝑊 = (𝐵 × 𝑆)
erler.q	⊢ ∼ = (𝑅 ~RL 𝑆)
erler.r	⊢ (𝜑 → 𝑅 ∈ CRing)
erler.s	⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))

Assertion

Ref	Expression
erler	⊢ (𝜑 → ∼ Er 𝑊)

Proof of Theorem erler

Dummy variables 𝑎 𝑏 𝑡 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2725	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
2	1	relopabiv 5822	. . . 4 ⊢ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
4		erler.q	. . . . 5 ⊢ ∼ = (𝑅 ~RL 𝑆)
5		erler.1	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
6		erler.2	. . . . . 6 ⊢ 0 = (0g‘𝑅)
7		erler.4	. . . . . 6 ⊢ · = (.r‘𝑅)
8		erler.5	. . . . . 6 ⊢ − = (-g‘𝑅)
9		erler.w	. . . . . 6 ⊢ 𝑊 = (𝐵 × 𝑆)
10		erler.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
11		eqid 2725	. . . . . . . . 9 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
12	11, 5	mgpbas 20092	. . . . . . . 8 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
13	12	submss 18769	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
14	10, 13	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
15	5, 6, 7, 8, 9, 1, 14	erlval 33048	. . . . 5 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
16	4, 15	eqtrid 2777	. . . 4 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
17	16	releqd 5780	. . 3 ⊢ (𝜑 → (Rel ∼ ↔ Rel {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑊 ∧ 𝑏 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}))
18	3, 17	mpbird 256	. 2 ⊢ (𝜑 → Rel ∼ )
19		simpl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑎 = 𝑥)
20	19	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (1st ‘𝑎) = (1st ‘𝑥))
21		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → 𝑏 = 𝑦)
22	21	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (2nd ‘𝑏) = (2nd ‘𝑦))
23	20, 22	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑥) · (2nd ‘𝑦)))
24	21	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (1st ‘𝑏) = (1st ‘𝑦))
25	19	fveq2d 6900	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (2nd ‘𝑎) = (2nd ‘𝑥))
26	24, 25	oveq12d 7437	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑦) · (2nd ‘𝑥)))
27	23, 26	oveq12d 7437	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))
28	27	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))
29	28	eqeq1d 2727	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ))
30	29	rexbidv 3168	. . . . . . . . 9 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ))
31	30	adantl 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑥 ∧ 𝑏 = 𝑦)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ))
32	16, 31	brab2d 32476	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∼ 𝑦 ↔ ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 )))
33	32	biimpa 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → ((𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ))
34	33	simplrd 768	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → 𝑦 ∈ 𝑊)
35	33	simplld 766	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → 𝑥 ∈ 𝑊)
36	34, 35	jca 510	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → (𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
37	33	simprd 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 )
38		erler.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ CRing)
39	38	crngringd 20198	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑅 ∈ Ring)
40	39	ringgrpd 20194	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ Grp)
41	40	ad3antrrr 728	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑅 ∈ Grp)
42	39	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑅 ∈ Ring)
43	35	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑥 ∈ 𝑊)
44		xp1st 8026	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐵 × 𝑆) → (1st ‘𝑥) ∈ 𝐵)
45	44, 9	eleq2s 2843	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑊 → (1st ‘𝑥) ∈ 𝐵)
46	43, 45	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (1st ‘𝑥) ∈ 𝐵)
47	14	ad3antrrr 728	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑆 ⊆ 𝐵)
48	34	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑦 ∈ 𝑊)
49		xp2nd 8027	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝐵 × 𝑆) → (2nd ‘𝑦) ∈ 𝑆)
50	49, 9	eleq2s 2843	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑊 → (2nd ‘𝑦) ∈ 𝑆)
51	48, 50	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (2nd ‘𝑦) ∈ 𝑆)
52	47, 51	sseldd 3977	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (2nd ‘𝑦) ∈ 𝐵)
53	5, 7, 42, 46, 52	ringcld 20211	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ((1st ‘𝑥) · (2nd ‘𝑦)) ∈ 𝐵)
54		xp1st 8026	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝐵 × 𝑆) → (1st ‘𝑦) ∈ 𝐵)
55	54, 9	eleq2s 2843	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑊 → (1st ‘𝑦) ∈ 𝐵)
56	48, 55	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (1st ‘𝑦) ∈ 𝐵)
57		xp2nd 8027	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐵 × 𝑆) → (2nd ‘𝑥) ∈ 𝑆)
58	57, 9	eleq2s 2843	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑊 → (2nd ‘𝑥) ∈ 𝑆)
59	43, 58	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (2nd ‘𝑥) ∈ 𝑆)
60	47, 59	sseldd 3977	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (2nd ‘𝑥) ∈ 𝐵)
61	5, 7, 42, 56, 60	ringcld 20211	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ((1st ‘𝑦) · (2nd ‘𝑥)) ∈ 𝐵)
62		eqid 2725	. . . . . . . . . . 11 ⊢ (invg‘𝑅) = (invg‘𝑅)
63	5, 8, 62	grpinvsub 18986	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘𝑥) · (2nd ‘𝑦)) ∈ 𝐵 ∧ ((1st ‘𝑦) · (2nd ‘𝑥)) ∈ 𝐵) → ((invg‘𝑅)‘(((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦))))
64	41, 53, 61, 63	syl3anc 1368	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ((invg‘𝑅)‘(((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦))))
65	64	oveq2d 7435	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (𝑡 · ((invg‘𝑅)‘(((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))))
66		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑡 ∈ 𝑆)
67	47, 66	sseldd 3977	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑡 ∈ 𝐵)
68	5, 8	grpsubcl 18984	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘𝑥) · (2nd ‘𝑦)) ∈ 𝐵 ∧ ((1st ‘𝑦) · (2nd ‘𝑥)) ∈ 𝐵) → (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))) ∈ 𝐵)
69	41, 53, 61, 68	syl3anc 1368	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))) ∈ 𝐵)
70	5, 7, 62, 42, 67, 69	ringmneg2 20253	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (𝑡 · ((invg‘𝑅)‘(((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = ((invg‘𝑅)‘(𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))))
71		simpr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 )
72	71	fveq2d 6900	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ((invg‘𝑅)‘(𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = ((invg‘𝑅)‘ 0 ))
73	6, 62	grpinvid 18964	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → ((invg‘𝑅)‘ 0 ) = 0 )
74	41, 73	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ((invg‘𝑅)‘ 0 ) = 0 )
75	70, 72, 74	3eqtrd 2769	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (𝑡 · ((invg‘𝑅)‘(((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = 0 )
76	65, 75	eqtr3d 2767	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 )
77	76	ex 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑡 ∈ 𝑆) → ((𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 → (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
78	77	reximdva 3157	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
79	37, 78	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 )
80	36, 79	jca 510	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
81		simpl 481	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → 𝑎 = 𝑦)
82	81	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (1st ‘𝑎) = (1st ‘𝑦))
83		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → 𝑏 = 𝑥)
84	83	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (2nd ‘𝑏) = (2nd ‘𝑥))
85	82, 84	oveq12d 7437	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑦) · (2nd ‘𝑥)))
86	83	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (1st ‘𝑏) = (1st ‘𝑥))
87	81	fveq2d 6900	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (2nd ‘𝑎) = (2nd ‘𝑦))
88	86, 87	oveq12d 7437	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑥) · (2nd ‘𝑦)))
89	85, 88	oveq12d 7437	. . . . . . . . 9 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦))))
90	89	oveq2d 7435	. . . . . . . 8 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))))
91	90	eqeq1d 2727	. . . . . . 7 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
92	91	rexbidv 3168	. . . . . 6 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
93	92	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝑦 ∧ 𝑏 = 𝑥)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 ))
94	16, 93	brab2d 32476	. . . 4 ⊢ (𝜑 → (𝑦 ∼ 𝑥 ↔ ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 )))
95	94	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → (𝑦 ∼ 𝑥 ↔ ((𝑦 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑦)))) = 0 )))
96	80, 95	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑥 ∼ 𝑦) → 𝑦 ∼ 𝑥)
97	10	ad6antr 734	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
98	97, 13	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑆 ⊆ 𝐵)
99	35	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → 𝑥 ∈ 𝑊)
100	99	ad4antr 730	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑥 ∈ 𝑊)
101	100, 9	eleqtrdi 2835	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑥 ∈ (𝐵 × 𝑆))
102		1st2nd2 8033	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 × 𝑆) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
103	101, 102	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
104		simpl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → 𝑎 = 𝑦)
105	104	fveq2d 6900	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (1st ‘𝑎) = (1st ‘𝑦))
106		simpr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → 𝑏 = 𝑧)
107	106	fveq2d 6900	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (2nd ‘𝑏) = (2nd ‘𝑧))
108	105, 107	oveq12d 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑦) · (2nd ‘𝑧)))
109	106	fveq2d 6900	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (1st ‘𝑏) = (1st ‘𝑧))
110	104	fveq2d 6900	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (2nd ‘𝑎) = (2nd ‘𝑦))
111	109, 110	oveq12d 7437	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑧) · (2nd ‘𝑦)))
112	108, 111	oveq12d 7437	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))
113	112	oveq2d 7435	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))
114	113	eqeq1d 2727	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
115	114	rexbidv 3168	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
116		oveq1 7426	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑢 → (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))
117	116	eqeq1d 2727	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑢 → ((𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ↔ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
118	117	cbvrexvw 3225	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ↔ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 )
119	115, 118	bitrdi 286	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑧) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
120	119	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑎 = 𝑦 ∧ 𝑏 = 𝑧)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
121	16, 120	brab2d 32476	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∼ 𝑧 ↔ ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊) ∧ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 )))
122	121	biimpa 475	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∼ 𝑧) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊) ∧ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
123	122	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → ((𝑦 ∈ 𝑊 ∧ 𝑧 ∈ 𝑊) ∧ ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ))
124	123	simplrd 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → 𝑧 ∈ 𝑊)
125	124	ad4antr 730	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑧 ∈ 𝑊)
126	125, 9	eleqtrdi 2835	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑧 ∈ (𝐵 × 𝑆))
127		1st2nd2 8033	. . . . . . 7 ⊢ (𝑧 ∈ (𝐵 × 𝑆) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
128	126, 127	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
129	100, 45	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (1st ‘𝑥) ∈ 𝐵)
130		xp1st 8026	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐵 × 𝑆) → (1st ‘𝑧) ∈ 𝐵)
131	130, 9	eleq2s 2843	. . . . . . 7 ⊢ (𝑧 ∈ 𝑊 → (1st ‘𝑧) ∈ 𝐵)
132	125, 131	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (1st ‘𝑧) ∈ 𝐵)
133	100, 58	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑥) ∈ 𝑆)
134		xp2nd 8027	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐵 × 𝑆) → (2nd ‘𝑧) ∈ 𝑆)
135	134, 9	eleq2s 2843	. . . . . . 7 ⊢ (𝑧 ∈ 𝑊 → (2nd ‘𝑧) ∈ 𝑆)
136	125, 135	syl 17	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑧) ∈ 𝑆)
137		simp-4r 782	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑡 ∈ 𝑆)
138		simplr 767	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑢 ∈ 𝑆)
139	11, 7	mgpplusg 20090	. . . . . . . . 9 ⊢ · = (+g‘(mulGrp‘𝑅))
140	139	submcl 18772	. . . . . . . 8 ⊢ ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ 𝑡 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆) → (𝑡 · 𝑢) ∈ 𝑆)
141	97, 137, 138, 140	syl3anc 1368	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑡 · 𝑢) ∈ 𝑆)
142	34	ad5antr 732	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑦 ∈ 𝑊)
143	142, 50	syl 17	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑦) ∈ 𝑆)
144	139	submcl 18772	. . . . . . 7 ⊢ ((𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) ∧ (𝑡 · 𝑢) ∈ 𝑆 ∧ (2nd ‘𝑦) ∈ 𝑆) → ((𝑡 · 𝑢) · (2nd ‘𝑦)) ∈ 𝑆)
145	97, 141, 143, 144	syl3anc 1368	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · 𝑢) · (2nd ‘𝑦)) ∈ 𝑆)
146	39	ad6antr 734	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑅 ∈ Ring)
147	98, 143	sseldd 3977	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑦) ∈ 𝐵)
148	98, 136	sseldd 3977	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑧) ∈ 𝐵)
149	5, 7, 146, 129, 148	ringcld 20211	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑥) · (2nd ‘𝑧)) ∈ 𝐵)
150	98, 133	sseldd 3977	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (2nd ‘𝑥) ∈ 𝐵)
151	5, 7, 146, 132, 150	ringcld 20211	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑧) · (2nd ‘𝑥)) ∈ 𝐵)
152	5, 7, 8, 146, 147, 149, 151	ringsubdi 20255	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥)))) = (((2nd ‘𝑦) · ((1st ‘𝑥) · (2nd ‘𝑧))) − ((2nd ‘𝑦) · ((1st ‘𝑧) · (2nd ‘𝑥)))))
153	38	ad6antr 734	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑅 ∈ CRing)
154	5, 7, 153, 147, 129, 148	crng12d 20210	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · ((1st ‘𝑥) · (2nd ‘𝑧))) = ((1st ‘𝑥) · ((2nd ‘𝑦) · (2nd ‘𝑧))))
155	5, 7, 153, 147, 148	crngcomd 20207	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · (2nd ‘𝑧)) = ((2nd ‘𝑧) · (2nd ‘𝑦)))
156	155	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑥) · ((2nd ‘𝑦) · (2nd ‘𝑧))) = ((1st ‘𝑥) · ((2nd ‘𝑧) · (2nd ‘𝑦))))
157	5, 7, 153, 129, 148, 147	crng12d 20210	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑥) · ((2nd ‘𝑧) · (2nd ‘𝑦))) = ((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))))
158	154, 156, 157	3eqtrd 2769	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · ((1st ‘𝑥) · (2nd ‘𝑧))) = ((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))))
159	5, 7, 153, 147, 132, 150	crng12d 20210	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · ((1st ‘𝑧) · (2nd ‘𝑥))) = ((1st ‘𝑧) · ((2nd ‘𝑦) · (2nd ‘𝑥))))
160	5, 7, 153, 147, 150	crngcomd 20207	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · (2nd ‘𝑥)) = ((2nd ‘𝑥) · (2nd ‘𝑦)))
161	160	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑧) · ((2nd ‘𝑦) · (2nd ‘𝑥))) = ((1st ‘𝑧) · ((2nd ‘𝑥) · (2nd ‘𝑦))))
162	5, 7, 153, 132, 150, 147	crng12d 20210	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑧) · ((2nd ‘𝑥) · (2nd ‘𝑦))) = ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))
163	159, 161, 162	3eqtrd 2769	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · ((1st ‘𝑧) · (2nd ‘𝑥))) = ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))
164	158, 163	oveq12d 7437	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((2nd ‘𝑦) · ((1st ‘𝑥) · (2nd ‘𝑧))) − ((2nd ‘𝑦) · ((1st ‘𝑧) · (2nd ‘𝑥)))) = (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))))
165	5, 7, 146, 129, 147	ringcld 20211	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑥) · (2nd ‘𝑦)) ∈ 𝐵)
166	142, 55	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (1st ‘𝑦) ∈ 𝐵)
167	5, 7, 146, 166, 150	ringcld 20211	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · (2nd ‘𝑥)) ∈ 𝐵)
168	5, 7, 8, 146, 148, 165, 167	ringsubdi 20255	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑧) · ((1st ‘𝑦) · (2nd ‘𝑥)))))
169	5, 7, 146, 166, 148	ringcld 20211	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · (2nd ‘𝑧)) ∈ 𝐵)
170	5, 7, 146, 132, 147	ringcld 20211	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑧) · (2nd ‘𝑦)) ∈ 𝐵)
171	5, 7, 8, 146, 150, 169, 170	ringsubdi 20255	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = (((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))))
172	168, 171	oveq12d 7437	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))) = ((((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑧) · ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)(((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))))
173	5, 7, 153, 166, 148, 150	crng12d 20210	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) = ((2nd ‘𝑧) · ((1st ‘𝑦) · (2nd ‘𝑥))))
174	173	oveq2d 7435	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥)))) = (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑧) · ((1st ‘𝑦) · (2nd ‘𝑥)))))
175	5, 7, 153, 148, 150	crngcomd 20207	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑧) · (2nd ‘𝑥)) = ((2nd ‘𝑥) · (2nd ‘𝑧)))
176	175	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) = ((1st ‘𝑦) · ((2nd ‘𝑥) · (2nd ‘𝑧))))
177	5, 7, 153, 150, 166, 148	crng12d 20210	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))) = ((1st ‘𝑦) · ((2nd ‘𝑥) · (2nd ‘𝑧))))
178	176, 177	eqtr4d 2768	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) = ((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))))
179	178	oveq1d 7434	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))) = (((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))))
180	174, 179	oveq12d 7437	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))))(+g‘𝑅)(((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))) = ((((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑧) · ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)(((2nd ‘𝑥) · ((1st ‘𝑦) · (2nd ‘𝑧))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))))
181	40	ad6antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑅 ∈ Grp)
182	5, 7, 146, 148, 165	ringcld 20211	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) ∈ 𝐵)
183	5, 7, 146, 148, 150	ringcld 20211	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑧) · (2nd ‘𝑥)) ∈ 𝐵)
184	5, 7, 146, 166, 183	ringcld 20211	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) ∈ 𝐵)
185	5, 7, 146, 150, 170	ringcld 20211	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))) ∈ 𝐵)
186		eqid 2725	. . . . . . . . . . . . 13 ⊢ (+g‘𝑅) = (+g‘𝑅)
187	5, 186, 8	grpnpncan 18999	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ Grp ∧ (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) ∈ 𝐵 ∧ ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) ∈ 𝐵 ∧ ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))) ∈ 𝐵)) → ((((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))))(+g‘𝑅)(((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))) = (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))))
188	181, 182, 184, 185, 187	syl13anc 1369	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))))(+g‘𝑅)(((1st ‘𝑦) · ((2nd ‘𝑧) · (2nd ‘𝑥))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦))))) = (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))))
189	172, 180, 188	3eqtr2rd 2772	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((2nd ‘𝑧) · ((1st ‘𝑥) · (2nd ‘𝑦))) − ((2nd ‘𝑥) · ((1st ‘𝑧) · (2nd ‘𝑦)))) = (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))))
190	152, 164, 189	3eqtrd 2769	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑦) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥)))) = (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))))
191	190	oveq2d 7435	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · 𝑢) · ((2nd ‘𝑦) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥))))) = ((𝑡 · 𝑢) · (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
192	98, 141	sseldd 3977	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑡 · 𝑢) ∈ 𝐵)
193	5, 8	grpsubcl 18984	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘𝑥) · (2nd ‘𝑧)) ∈ 𝐵 ∧ ((1st ‘𝑧) · (2nd ‘𝑥)) ∈ 𝐵) → (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥))) ∈ 𝐵)
194	181, 149, 151, 193	syl3anc 1368	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥))) ∈ 𝐵)
195	5, 7, 146, 192, 147, 194	ringassd 20209	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · 𝑢) · (2nd ‘𝑦)) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥)))) = ((𝑡 · 𝑢) · ((2nd ‘𝑦) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥))))))
196	98, 138	sseldd 3977	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑢 ∈ 𝐵)
197	98, 137	sseldd 3977	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑡 ∈ 𝐵)
198	5, 7, 153, 196, 148, 197	cringmul32d 33028	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · 𝑡) = ((𝑢 · 𝑡) · (2nd ‘𝑧)))
199	5, 7, 153, 196, 197	crngcomd 20207	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑢 · 𝑡) = (𝑡 · 𝑢))
200	199	oveq1d 7434	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · 𝑡) · (2nd ‘𝑧)) = ((𝑡 · 𝑢) · (2nd ‘𝑧)))
201	198, 200	eqtrd 2765	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · 𝑡) = ((𝑡 · 𝑢) · (2nd ‘𝑧)))
202	201	oveq1d 7434	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑢 · (2nd ‘𝑧)) · 𝑡) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = (((𝑡 · 𝑢) · (2nd ‘𝑧)) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))
203	5, 7, 146, 196, 148	ringcld 20211	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑢 · (2nd ‘𝑧)) ∈ 𝐵)
204	181, 165, 167, 68	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))) ∈ 𝐵)
205	5, 7, 146, 203, 197, 204	ringassd 20209	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑢 · (2nd ‘𝑧)) · 𝑡) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = ((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))))
206	5, 7, 146, 192, 148, 204	ringassd 20209	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · 𝑢) · (2nd ‘𝑧)) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = ((𝑡 · 𝑢) · ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))))
207	202, 205, 206	3eqtr3d 2773	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = ((𝑡 · 𝑢) · ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))))
208	5, 7, 153, 197, 150, 196	cringmul32d 33028	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · (2nd ‘𝑥)) · 𝑢) = ((𝑡 · 𝑢) · (2nd ‘𝑥)))
209	208	oveq1d 7434	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · (2nd ‘𝑥)) · 𝑢) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = (((𝑡 · 𝑢) · (2nd ‘𝑥)) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))
210	5, 7, 146, 197, 150	ringcld 20211	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑡 · (2nd ‘𝑥)) ∈ 𝐵)
211	5, 8	grpsubcl 18984	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘𝑦) · (2nd ‘𝑧)) ∈ 𝐵 ∧ ((1st ‘𝑧) · (2nd ‘𝑦)) ∈ 𝐵) → (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))) ∈ 𝐵)
212	181, 169, 170, 211	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))) ∈ 𝐵)
213	5, 7, 146, 210, 196, 212	ringassd 20209	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · (2nd ‘𝑥)) · 𝑢) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = ((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))))
214	5, 7, 146, 192, 150, 212	ringassd 20209	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · 𝑢) · (2nd ‘𝑥)) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = ((𝑡 · 𝑢) · ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))))
215	209, 213, 214	3eqtr3d 2773	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))) = ((𝑡 · 𝑢) · ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))))
216	207, 215	oveq12d 7437	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))) = (((𝑡 · 𝑢) · ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · 𝑢) · ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
217	5, 7, 146, 148, 204	ringcld 20211	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) ∈ 𝐵)
218	5, 7, 146, 150, 212	ringcld 20211	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) ∈ 𝐵)
219	5, 186, 7	ringdi 20212	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ ((𝑡 · 𝑢) ∈ 𝐵 ∧ ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) ∈ 𝐵 ∧ ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) ∈ 𝐵)) → ((𝑡 · 𝑢) · (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))) = (((𝑡 · 𝑢) · ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · 𝑢) · ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
220	146, 192, 217, 218, 219	syl13anc 1369	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · 𝑢) · (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))) = (((𝑡 · 𝑢) · ((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · 𝑢) · ((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
221	216, 220	eqtr4d 2768	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))) = ((𝑡 · 𝑢) · (((2nd ‘𝑧) · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))(+g‘𝑅)((2nd ‘𝑥) · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
222	191, 195, 221	3eqtr4d 2775	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · 𝑢) · (2nd ‘𝑦)) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥)))) = (((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))))
223		simpllr 774	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 )
224	223	oveq2d 7435	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = ((𝑢 · (2nd ‘𝑧)) · 0 ))
225	5, 7, 6, 146, 203	ringrzd 20244	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · 0 ) = 0 )
226	224, 225	eqtrd 2765	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥))))) = 0 )
227		simpr 483	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 )
228	227	oveq2d 7435	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))) = ((𝑡 · (2nd ‘𝑥)) · 0 ))
229	5, 7, 6, 146, 210	ringrzd 20244	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · (2nd ‘𝑥)) · 0 ) = 0 )
230	228, 229	eqtrd 2765	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦))))) = 0 )
231	226, 230	oveq12d 7437	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑢 · (2nd ‘𝑧)) · (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))))(+g‘𝑅)((𝑡 · (2nd ‘𝑥)) · (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))))) = ( 0 (+g‘𝑅) 0 ))
232	5, 6	grpidcl 18930	. . . . . . . . 9 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵)
233	181, 232	syl 17	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 0 ∈ 𝐵)
234	5, 186, 6, 181, 233	grplidd 18934	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → ( 0 (+g‘𝑅) 0 ) = 0 )
235	222, 231, 234	3eqtrd 2769	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → (((𝑡 · 𝑢) · (2nd ‘𝑦)) · (((1st ‘𝑥) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑥)))) = 0 )
236	5, 4, 98, 6, 7, 8, 103, 128, 129, 132, 133, 136, 145, 235	erlbrd 33053	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) ∧ 𝑢 ∈ 𝑆) ∧ (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 ) → 𝑥 ∼ 𝑧)
237	123	simprd 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 )
238	237	ad2antrr 724	. . . . 5 ⊢ (((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → ∃𝑢 ∈ 𝑆 (𝑢 · (((1st ‘𝑦) · (2nd ‘𝑧)) − ((1st ‘𝑧) · (2nd ‘𝑦)))) = 0 )
239	236, 238	r19.29a 3151	. . . 4 ⊢ (((((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 ) → 𝑥 ∼ 𝑧)
240	37	adantr 479	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑦)) − ((1st ‘𝑦) · (2nd ‘𝑥)))) = 0 )
241	239, 240	r19.29a 3151	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∼ 𝑦) ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)
242	241	anasss 465	. 2 ⊢ ((𝜑 ∧ (𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧)) → 𝑥 ∼ 𝑧)
243		erler.3	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝑅)
244	11, 243	ringidval 20135	. . . . . . . . . 10 ⊢ 1 = (0g‘(mulGrp‘𝑅))
245	244	subm0cl 18771	. . . . . . . . 9 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 1 ∈ 𝑆)
246	10, 245	syl 17	. . . . . . . 8 ⊢ (𝜑 → 1 ∈ 𝑆)
247	246	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 1 ∈ 𝑆)
248		oveq1 7426	. . . . . . . . 9 ⊢ (𝑡 = 1 → (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = ( 1 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))))
249	248	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑡 = 1 → ((𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ↔ ( 1 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
250	249	adantl 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑊) ∧ 𝑡 = 1 ) → ((𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ↔ ( 1 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
251	39	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑅 ∈ Ring)
252	45	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (1st ‘𝑥) ∈ 𝐵)
253	14	adantr 479	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → 𝑆 ⊆ 𝐵)
254	58	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (2nd ‘𝑥) ∈ 𝑆)
255	253, 254	sseldd 3977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (2nd ‘𝑥) ∈ 𝐵)
256	5, 7, 251, 252, 255	ringcld 20211	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ((1st ‘𝑥) · (2nd ‘𝑥)) ∈ 𝐵)
257	5, 6, 8	grpsubid 18988	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Grp ∧ ((1st ‘𝑥) · (2nd ‘𝑥)) ∈ 𝐵) → (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥))) = 0 )
258	40, 256, 257	syl2an2r 683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥))) = 0 )
259	258	oveq2d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ( 1 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = ( 1 · 0 ))
260	40, 232	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ 𝐵)
261	5, 7, 243, 39, 260	ringlidmd 20220	. . . . . . . . 9 ⊢ (𝜑 → ( 1 · 0 ) = 0 )
262	261	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ( 1 · 0 ) = 0 )
263	259, 262	eqtrd 2765	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ( 1 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 )
264	247, 250, 263	rspcedvd 3608	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑊) → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 )
265	264	ex 411	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑊 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
266	265	pm4.71d 560	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 )))
267		pm4.24 562	. . . . 5 ⊢ (𝑥 ∈ 𝑊 ↔ (𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
268	267	anbi1i 622	. . . 4 ⊢ ((𝑥 ∈ 𝑊 ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ) ↔ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
269	266, 268	bitrdi 286	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑊 ↔ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 )))
270		simpl 481	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → 𝑎 = 𝑥)
271	270	fveq2d 6900	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (1st ‘𝑎) = (1st ‘𝑥))
272		simpr 483	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → 𝑏 = 𝑥)
273	272	fveq2d 6900	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (2nd ‘𝑏) = (2nd ‘𝑥))
274	271, 273	oveq12d 7437	. . . . . . . . 9 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑥) · (2nd ‘𝑥)))
275	272	fveq2d 6900	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (1st ‘𝑏) = (1st ‘𝑥))
276	270	fveq2d 6900	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (2nd ‘𝑎) = (2nd ‘𝑥))
277	275, 276	oveq12d 7437	. . . . . . . . 9 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑥) · (2nd ‘𝑥)))
278	274, 277	oveq12d 7437	. . . . . . . 8 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥))))
279	278	oveq2d 7435	. . . . . . 7 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))))
280	279	eqeq1d 2727	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
281	280	rexbidv 3168	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑥) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
282	281	adantl 480	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝑥 ∧ 𝑏 = 𝑥)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 ))
283	16, 282	brab2d 32476	. . 3 ⊢ (𝜑 → (𝑥 ∼ 𝑥 ↔ ((𝑥 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑥) · (2nd ‘𝑥)) − ((1st ‘𝑥) · (2nd ‘𝑥)))) = 0 )))
284	269, 283	bitr4d 281	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥))
285	18, 96, 242, 284	iserd 8751	1 ⊢ (𝜑 → ∼ Er 𝑊)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃wrex 3059   ⊆ wss 3944  ⟨cop 4636   class class class wbr 5149  {copab 5211   × cxp 5676  Rel wrel 5683  ‘cfv 6549  (class class class)co 7419  1^st c1st 7992  2^nd c2nd 7993   Er wer 8722  Basecbs 17183  +_gcplusg 17236  ._rcmulr 17237  0_gc0g 17424  SubMndcsubmnd 18742  Grpcgrp 18898  inv_gcminusg 18899  -_gcsg 18900  mulGrpcmgp 20086  1_rcur 20133  Ringcrg 20185  CRingccrg 20186   ~_RL cerl 33043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18744  df-grp 18901  df-minusg 18902  df-sbg 18903  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-erl 33045

This theorem is referenced by:  rlocaddval  33058  rlocmulval  33059  rloccring  33060  rlocf1  33063  fracfld  33094  zringfrac  33369