Theorem rloccring

Description: The ring localization L of a commutative ring R by a multiplicatively closed set S is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025)

	Ref	Expression
Hypotheses	rlocaddval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
	rlocaddval.2	⊢ · = ( .r ‘ 𝑅 )
	rlocaddval.3	⊢ + = ( +g ‘ 𝑅 )
	rlocaddval.4	⊢ 𝐿 = ( 𝑅 RLocal 𝑆 )
	rlocaddval.5	⊢ ∼ = ( 𝑅 ~RL 𝑆 )
	rlocaddval.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	rlocaddval.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
Assertion	rloccring	⊢ ( 𝜑 → 𝐿 ∈ CRing )

Proof

Step	Hyp	Ref	Expression
1		rlocaddval.1	⊢ 𝐵 = ( Base ‘ 𝑅 )
2		rlocaddval.2	⊢ · = ( .r ‘ 𝑅 )
3		rlocaddval.3	⊢ + = ( +g ‘ 𝑅 )
4		rlocaddval.4	⊢ 𝐿 = ( 𝑅 RLocal 𝑆 )
5		rlocaddval.5	⊢ ∼ = ( 𝑅 ~RL 𝑆 )
6		rlocaddval.r	⊢ ( 𝜑 → 𝑅 ∈ CRing )
7		rlocaddval.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
8		eqid	⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
9		eqid	⊢ ( -g ‘ 𝑅 ) = ( -g ‘ 𝑅 )
10		eqid	⊢ ( 𝐵 × 𝑆 ) = ( 𝐵 × 𝑆 )
11		eqid	⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
12	11 1	mgpbas	⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
13	12	submss	⊢ ( 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) → 𝑆 ⊆ 𝐵 )
14	7 13	syl	⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
15	1 8 2 9 10 4 5 6 14	rlocbas	⊢ ( 𝜑 → ( ( 𝐵 × 𝑆 ) / ∼ ) = ( Base ‘ 𝐿 ) )
16		eqidd	⊢ ( 𝜑 → ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) )
17		eqidd	⊢ ( 𝜑 → ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 ) )
18		eqid	⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
19		eqid	⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 )
20		eqid	⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 )
21	6	crngringd	⊢ ( 𝜑 → 𝑅 ∈ Ring )
22	1 8	ring0cl	⊢ ( 𝑅 ∈ Ring → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
23	21 22	syl	⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
24		eqid	⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
25	11 24	ringidval	⊢ ( 1r ‘ 𝑅 ) = ( 0g ‘ ( mulGrp ‘ 𝑅 ) )
26	25	subm0cl	⊢ ( 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
27	7 26	syl	⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
28	23 27	opelxpd	⊢ ( 𝜑 → ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
29	5	ovexi	⊢ ∼ ∈ V
30	29	ecelqsi	⊢ ( ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
31	28 30	syl	⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
32	31 15	eleqtrd	⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( Base ‘ 𝐿 ) )
33	15	eleq2d	⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ↔ 𝑥 ∈ ( Base ‘ 𝐿 ) ) )
34	33	biimpar	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
35		simpr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
36	35	oveq2d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
37	21	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
38	7	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
39	38 13	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
40		simplr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
41	39 40	sseldd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
42	1 2 8 37 41	ringlzd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · 𝑏 ) = ( 0g ‘ 𝑅 ) )
43		simpllr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
44	1 2 24 37 43	ringridmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑎 · ( 1r ‘ 𝑅 ) ) = 𝑎 )
45	42 44	oveq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) = ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) 𝑎 ) )
46		eqid	⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
47	37	ringgrpd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
48	1 46 8 47 43	grplidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) 𝑎 ) = 𝑎 )
49	45 48	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) = 𝑎 )
50	1 2 24 37 41	ringlidmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 1r ‘ 𝑅 ) · 𝑏 ) = 𝑏 )
51	49 50	opeq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ = ⟨ 𝑎 , 𝑏 ⟩ )
52	51	eceq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
53	6	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
54	23	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
55	38 26	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
56	1 2 46 4 5 53 38 54 43 55 40 20	rlocaddval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ )
57	52 56 35	3eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = 𝑥 )
58	36 57	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
59		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
60	59	elrlocbasi	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
61	58 60	r19.29vva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
62	34 61	syldan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
63	1 2 3 4 5 53 38 43 54 40 55 20	rlocaddval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
64	35	oveq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
65	44 42	oveq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) = ( 𝑎 + ( 0g ‘ 𝑅 ) ) )
66	1 3 8 47 43	grpridd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑎 + ( 0g ‘ 𝑅 ) ) = 𝑎 )
67	65 66	eqtr2d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 = ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) )
68	1 2 24 37 41	ringridmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · ( 1r ‘ 𝑅 ) ) = 𝑏 )
69	68	eqcomd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 = ( 𝑏 · ( 1r ‘ 𝑅 ) ) )
70	67 69	opeq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ )
71	70	eceq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
72	35 71	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
73	63 64 72	3eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
74	73 60	r19.29vva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
75	34 74	syldan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
76	18 19 20 32 62 75	ismgmid2	⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ = ( 0g ‘ 𝐿 ) )
77		simp-4r	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
78		simpr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
79	77 78	oveq12d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
80	6	ad8antr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
81	7	ad8antr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
82		simp-6r	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
83		simpllr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑐 ∈ 𝐵 )
84		simp-5r	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
85		simplr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑑 ∈ 𝑆 )
86	1 2 3 4 5 80 81 82 83 84 85 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
87	80	crnggrpd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
88	21	ad8antr	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
89	81 13	syl	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
90	89 85	sseldd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑑 ∈ 𝐵 )
91	1 2 88 82 90	ringcld	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑑 ) ∈ 𝐵 )
92	89 84	sseldd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
93	1 2 88 83 92	ringcld	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑐 · 𝑏 ) ∈ 𝐵 )
94	1 3 87 91 93	grpcld	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) ∈ 𝐵 )
95	11 2	mgpplusg	⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
96	95 81 84 85	submcld	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝑆 )
97	94 96	opelxpd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
98	29	ecelqsi	⊢ ( ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
99	97 98	syl	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
100	86 99	eqeltrd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
101	79 100	eqeltrd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
102		simp-4r	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
103	102	elrlocbasi	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑐 ∈ 𝐵 ∃ 𝑑 ∈ 𝑆 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
104	101 103	r19.29vva	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
105		simplr	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
106	105	elrlocbasi	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
107	104 106	r19.29vva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
108	107	3impa	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
109	6	ad10antr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
110	109	crnggrpd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
111	21	ad10antr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
112		simp-9r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
113	7	ad10antr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
114	113 13	syl	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
115		simp-5r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑑 ∈ 𝑆 )
116	114 115	sseldd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑑 ∈ 𝐵 )
117	1 2 111 112 116	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑑 ) ∈ 𝐵 )
118		simplr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑓 ∈ 𝑆 )
119	114 118	sseldd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑓 ∈ 𝐵 )
120	1 2 111 117 119	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑓 ) ∈ 𝐵 )
121		simp-6r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑐 ∈ 𝐵 )
122		simp-8r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
123	114 122	sseldd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
124	1 2 111 121 123	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑏 ) ∈ 𝐵 )
125	1 2 111 124 119	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑓 ) ∈ 𝐵 )
126		simpllr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑒 ∈ 𝐵 )
127	1 2 111 123 116	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝐵 )
128	1 2 111 126 127	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) ∈ 𝐵 )
129	1 3 110 120 125 128	grpassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) ) )
130	1 2 111 112 116 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑓 ) = ( 𝑎 · ( 𝑑 · 𝑓 ) ) )
131	1 2 109 121 123 119	crng32d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑓 ) = ( ( 𝑐 · 𝑓 ) · 𝑏 ) )
132	1 2 109 126 123 116	crng12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) = ( 𝑏 · ( 𝑒 · 𝑑 ) ) )
133	1 2 111 126 116	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · 𝑑 ) ∈ 𝐵 )
134	1 2 109 123 133	crngcomd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑒 · 𝑑 ) ) = ( ( 𝑒 · 𝑑 ) · 𝑏 ) )
135	132 134	eqtrd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) = ( ( 𝑒 · 𝑑 ) · 𝑏 ) )
136	131 135	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) )
137	130 136	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
138	129 137	eqtrd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
139	1 3 2 111 117 124 119	ringdird	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) = ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) )
140	139	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) )
141	1 2 111 121 119	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑓 ) ∈ 𝐵 )
142	1 3 2 111 141 133 123	ringdird	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) = ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) )
143	142	oveq2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
144	138 140 143	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) )
145	1 2 111 123 116 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( 𝑏 · ( 𝑑 · 𝑓 ) ) )
146	144 145	opeq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
147	146	eceq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
148	1 3 110 117 124	grpcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) ∈ 𝐵 )
149	95 113 122 115	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝑆 )
150	1 2 3 4 5 109 113 148 126 149 118 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
151	1 3 110 141 133	grpcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ∈ 𝐵 )
152	95 113 115 118	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑑 · 𝑓 ) ∈ 𝑆 )
153	1 2 3 4 5 109 113 112 151 122 152 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
154	147 150 153	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
155	1 2 3 4 5 109 113 112 121 122 115 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
156	155	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
157	1 2 3 4 5 109 113 121 126 115 118 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ )
158	157	oveq2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
159	154 156 158	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
160		simp-7r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
161		simp-4r	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
162	160 161	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
163		simpr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ )
164	162 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
165	161 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
166	160 165	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
167	159 164 166	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
168		simpr3	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
169	168	ad6antr	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
170	169	elrlocbasi	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝑆 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ )
171	167 170	r19.29vva	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
172		simpr2	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
173	172	ad3antrrr	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
174	173	elrlocbasi	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑐 ∈ 𝐵 ∃ 𝑑 ∈ 𝑆 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
175	171 174	r19.29vva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
176		simpr1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
177	176	elrlocbasi	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
178	175 177	r19.29vva	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
179	15 16 108 178 31 61 74	ismndd	⊢ ( 𝜑 → 𝐿 ∈ Mnd )
180		eqid	⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
181	1 180 47 43	grpinvcld	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) ∈ 𝐵 )
182	181 40	opelxpd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ∈ ( 𝐵 × 𝑆 ) )
183	29	ecelqsi	⊢ ( ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
184	182 183	syl	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
185		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ )
186		simplr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
187	185 186	oveq12d	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
188	187	eqeq1d	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → ( ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ↔ ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
189	1 2 3 4 5 53 38 181 43 40 40 20	rlocaddval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ )
190	1 3 8 180 47 43	grplinvd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) = ( 0g ‘ 𝑅 ) )
191	190	oveq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) · 𝑏 ) = ( ( 0g ‘ 𝑅 ) · 𝑏 ) )
192	1 3 2 37 181 43 41	ringdird	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) · 𝑏 ) = ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) )
193	191 192 42	3eqtr3d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) = ( 0g ‘ 𝑅 ) )
194	193	opeq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ )
195	194	eceq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ = [ ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ )
196	1 8 24 2 9 10 5 6 7	erler	⊢ ( 𝜑 → ∼ Er ( 𝐵 × 𝑆 ) )
197	196	ad4antr	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∼ Er ( 𝐵 × 𝑆 ) )
198		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ )
199		eqidd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ )
200	95 38 40 40	submcld	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · 𝑏 ) ∈ 𝑆 )
201	1 2 24 37 54	ringridmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
202	39 200	sseldd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · 𝑏 ) ∈ 𝐵 )
203	1 2 8 37 202	ringlzd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) = ( 0g ‘ 𝑅 ) )
204	201 203	oveq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) = ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) )
205	1 8 9	grpsubid	⊢ ( ( 𝑅 ∈ Grp ∧ ( 0g ‘ 𝑅 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
206	47 54 205	syl2anc	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
207	204 206	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) = ( 0g ‘ 𝑅 ) )
208	207	oveq2d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) ) = ( ( 𝑏 · 𝑏 ) · ( 0g ‘ 𝑅 ) ) )
209	1 2 8 37 202	ringrzd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
210	208 209	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) ) = ( 0g ‘ 𝑅 ) )
211	1 5 39 8 2 9 198 199 54 54 200 55 200 210	erlbrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ∼ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ )
212	197 211	erthi	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
213	189 195 212	3eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
214	184 188 213	rspcedvd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑢 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
215	214 60	r19.29vva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑢 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
216	15 16 76 179 215	isgrpd2e	⊢ ( 𝜑 → 𝐿 ∈ Grp )
217	77 78	oveq12d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
218		eqid	⊢ ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 )
219	1 2 3 4 5 80 81 82 83 84 85 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
220	1 2 88 82 83	ringcld	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) ∈ 𝐵 )
221	220 96	opelxpd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
222	29	ecelqsi	⊢ ( ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
223	221 222	syl	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
224	219 223	eqeltrd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
225	217 224	eqeltrd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
226	225 103	r19.29vva	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
227	226 106	r19.29vva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
228	227	3impa	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
229	1 2 111 112 121 126	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑐 ) · 𝑒 ) = ( 𝑎 · ( 𝑐 · 𝑒 ) ) )
230	229 145	opeq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
231	230	eceq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
232	1 2 111 112 121	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) ∈ 𝐵 )
233	1 2 3 4 5 109 113 232 126 149 118 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
234	1 2 111 121 126	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑒 ) ∈ 𝐵 )
235	1 2 3 4 5 109 113 112 234 122 152 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
236	231 233 235	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
237	1 2 3 4 5 109 113 112 121 122 115 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
238	237	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
239	1 2 3 4 5 109 113 121 126 115 118 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ )
240	239	oveq2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
241	236 238 240	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
242	160 161	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
243	242 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
244	161 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
245	160 244	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
246	241 243 245	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
247	246 170	r19.29vva	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
248	247 174	r19.29vva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
249	248 177	r19.29vva	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
250	196	ad10antr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ∼ Er ( 𝐵 × 𝑆 ) )
251		eqidd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ = ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
252	1 2 111 112 123	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑏 ) ∈ 𝐵 )
253	1 3 2 111 252 141 133	ringdid	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑏 ) · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) )
254	1 2 111 112 123 151	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑏 ) · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
255	253 254	eqtr3d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
256	11	crngmgp	⊢ ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
257	6 256	syl	⊢ ( 𝜑 → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
258	257	ad10antr	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
259	12 95 258 112 121 123 119	cmn4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) = ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) )
260	12 95 258 112 126 123 116	cmn4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) = ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) )
261	259 260	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) )
262	1 2 109 123 112 151	crng12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
263	255 261 262	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) = ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
264	1 2 109 127 123 119	crng12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑏 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) )
265	145	oveq2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
266	264 265	eqtrd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
267	263 266	opeq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ = ⟨ ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) , ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) ⟩ )
268	1 2 111 112 151	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ∈ 𝐵 )
269	1 2 111 123 268	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) ∈ 𝐵 )
270	95 113 122 152	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑑 · 𝑓 ) ) ∈ 𝑆 )
271	95 113 122 270	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) ∈ 𝑆 )
272		eqidd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) = ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
273		eqidd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
274	1 5 109 113 2 251 267 268 269 270 271 122 272 273	erlbr2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ∼ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ )
275	250 274	erthi	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ = [ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ ] ∼ )
276	1 2 3 4 5 109 113 112 151 122 152 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
277	1 2 111 112 126	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑒 ) ∈ 𝐵 )
278	95 113 122 118	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑓 ) ∈ 𝑆 )
279	1 2 3 4 5 109 113 232 277 149 278 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ ] ∼ )
280	275 276 279	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) )
281	157	oveq2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
282	1 2 3 4 5 109 113 112 126 122 118 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ )
283	237 282	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) )
284	280 281 283	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
285	160 165	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
286	160 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
287	242 286	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
288	284 285 287	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
289	288 170	r19.29vva	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
290	289 174	r19.29vva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
291	290 177	r19.29vva	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
292	1 3 2 111 117 124 126	ringdird	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) = ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) )
293	292	opeq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ )
294	1 2 111 117 126 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) · 𝑓 ) = ( ( 𝑎 · 𝑑 ) · ( 𝑒 · 𝑓 ) ) )
295	1 2 111 117 126	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑒 ) ∈ 𝐵 )
296	1 2 109 119 295	crngcomd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) = ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) · 𝑓 ) )
297	12 95 258 112 126 116 119	cmn4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) = ( ( 𝑎 · 𝑑 ) · ( 𝑒 · 𝑓 ) ) )
298	294 296 297	3eqtr4rd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) = ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) )
299	1 2 111 124 126 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑏 ) · 𝑒 ) · 𝑓 ) = ( ( 𝑐 · 𝑏 ) · ( 𝑒 · 𝑓 ) ) )
300	1 2 111 124 126	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑒 ) ∈ 𝐵 )
301	1 2 109 119 300	crngcomd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) = ( ( ( 𝑐 · 𝑏 ) · 𝑒 ) · 𝑓 ) )
302	12 95 258 121 126 123 119	cmn4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) = ( ( 𝑐 · 𝑏 ) · ( 𝑒 · 𝑓 ) ) )
303	299 301 302	3eqtr4rd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) )
304	298 303	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) = ( ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) + ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
305	1 3 2 111 119 295 300	ringdid	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) = ( ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) + ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
306	304 305	eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) = ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
307	114 278	sseldd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑓 ) ∈ 𝐵 )
308	1 2 111 116 307 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑑 · ( 𝑏 · 𝑓 ) ) · 𝑓 ) = ( 𝑑 · ( ( 𝑏 · 𝑓 ) · 𝑓 ) ) )
309	1 2 109 123 116	crngcomd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) = ( 𝑑 · 𝑏 ) )
310	309	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( ( 𝑑 · 𝑏 ) · 𝑓 ) )
311	1 2 111 116 123 119	ringassd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑑 · 𝑏 ) · 𝑓 ) = ( 𝑑 · ( 𝑏 · 𝑓 ) ) )
312	310 311	eqtrd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( 𝑑 · ( 𝑏 · 𝑓 ) ) )
313	312	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) = ( ( 𝑑 · ( 𝑏 · 𝑓 ) ) · 𝑓 ) )
314	1 2 109 307 116 119	crng12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) = ( 𝑑 · ( ( 𝑏 · 𝑓 ) · 𝑓 ) ) )
315	308 313 314	3eqtr4rd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) = ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) )
316	306 315	opeq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ = ⟨ ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) , ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) ⟩ )
317	1 3 110 295 300	grpcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ∈ 𝐵 )
318	1 2 111 119 317	ringcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) ∈ 𝐵 )
319	145 270	eqeltrd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) ∈ 𝑆 )
320	95 113 319 118	submcld	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) ∈ 𝑆 )
321		eqidd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) = ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
322	114 319	sseldd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) ∈ 𝐵 )
323	1 2 109 322 119	crngcomd	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) = ( 𝑓 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) )
324	1 5 109 113 2 293 316 317 318 319 320 118 321 323	erlbr2d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ∼ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ )
325	250 324	erthi	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
326	1 2 3 4 5 109 113 148 126 149 118 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
327	1 2 3 4 5 109 113 277 234 278 152 20	rlocaddval	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
328	325 326 327	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
329	155	oveq1d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
330	282 239	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
331	328 329 330	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
332	162 163	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
333	286 244	oveq12d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
334	331 332 333	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
335	334 170	r19.29vva	⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
336	335 174	r19.29vva	⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
337	336 177	r19.29vva	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
338	14 27	sseldd	⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
339	338 27	opelxpd	⊢ ( 𝜑 → ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
340	29	ecelqsi	⊢ ( ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
341	339 340	syl	⊢ ( 𝜑 → [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
342	35	oveq2d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
343	1 2 24 37 43	ringlidmd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 1r ‘ 𝑅 ) · 𝑎 ) = 𝑎 )
344	343 50	opeq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ = ⟨ 𝑎 , 𝑏 ⟩ )
345	344	eceq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
346	39 55	sseldd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
347	1 2 3 4 5 53 38 346 43 55 40 218	rlocmulval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ )
348	345 347 35	3eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = 𝑥 )
349	342 348	eqtrd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = 𝑥 )
350	349 60	r19.29vva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = 𝑥 )
351	1 2 3 4 5 53 38 43 346 40 55 218	rlocmulval	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
352	35	oveq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
353	44	eqcomd	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 = ( 𝑎 · ( 1r ‘ 𝑅 ) ) )
354	353 69	opeq12d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ )
355	354	eceq1d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ = [ ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
356	351 352 355	3eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
357	356 35	eqtr4d	⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
358	357 60	r19.29vva	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
359	1 2 80 82 83	crngcomd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) = ( 𝑐 · 𝑎 ) )
360	1 2 80 92 90	crngcomd	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) = ( 𝑑 · 𝑏 ) )
361	359 360	opeq12d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ = ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ )
362	361	eceq1d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ = [ ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ ] ∼ )
363	1 2 3 4 5 80 81 83 82 85 84 218	rlocmulval	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ ] ∼ )
364	362 219 363	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
365	78 77	oveq12d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
366	364 217 365	3eqtr4d	⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
367	366 103	r19.29vva	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
368	367 106	r19.29vva	⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
369	368	3impa	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
370	15 16 17 216 228 249 291 337 341 350 358 369	iscrngd	⊢ ( 𝜑 → 𝐿 ∈ CRing )