Theorem erlbrd 33123

Description: Deduce the ring localization equivalence relation. If for some 𝑇 ∈ 𝑆 we have 𝑇 · (𝐸 · 𝐻 − 𝐹 · 𝐺) = 0, then pairs ⟨𝐸, 𝐺⟩ and ⟨𝐹, 𝐻⟩ are equivalent under the localization relation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
erlcl1.b	⊢ 𝐵 = (Base‘𝑅)
erlcl1.e	⊢ ∼ = (𝑅 ~RL 𝑆)
erlcl1.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
erldi.1	⊢ 0 = (0g‘𝑅)
erldi.2	⊢ · = (.r‘𝑅)
erldi.3	⊢ − = (-g‘𝑅)
erlbrd.u	⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)
erlbrd.v	⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)
erlbrd.e	⊢ (𝜑 → 𝐸 ∈ 𝐵)
erlbrd.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
erlbrd.g	⊢ (𝜑 → 𝐺 ∈ 𝑆)
erlbrd.h	⊢ (𝜑 → 𝐻 ∈ 𝑆)
erlbrd.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
erlbrd.2	⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )

Assertion

Ref	Expression
erlbrd	⊢ (𝜑 → 𝑈 ∼ 𝑉)

Proof of Theorem erlbrd

Dummy variables 𝑎 𝑏 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		erlbrd.u	. . . . 5 ⊢ (𝜑 → 𝑈 = ⟨𝐸, 𝐺⟩)
2		erlbrd.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐵)
3		erlbrd.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑆)
4	2, 3	opelxpd 5713	. . . . 5 ⊢ (𝜑 → ⟨𝐸, 𝐺⟩ ∈ (𝐵 × 𝑆))
5	1, 4	eqeltrd 2826	. . . 4 ⊢ (𝜑 → 𝑈 ∈ (𝐵 × 𝑆))
6		erlbrd.v	. . . . 5 ⊢ (𝜑 → 𝑉 = ⟨𝐹, 𝐻⟩)
7		erlbrd.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
8		erlbrd.h	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ 𝑆)
9	7, 8	opelxpd 5713	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐻⟩ ∈ (𝐵 × 𝑆))
10	6, 9	eqeltrd 2826	. . . 4 ⊢ (𝜑 → 𝑉 ∈ (𝐵 × 𝑆))
11	5, 10	jca 510	. . 3 ⊢ (𝜑 → (𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)))
12		erlbrd.1	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
13		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → 𝑡 = 𝑇)
14	13	oveq1d 7431	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
15	14	eqeq1d 2728	. . . 4 ⊢ ((𝜑 ∧ 𝑡 = 𝑇) → ((𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ↔ (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
16		erlbrd.2	. . . 4 ⊢ (𝜑 → (𝑇 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )
17	12, 15, 16	rspcedvd 3609	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )
18	11, 17	jca 510	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
19		erlcl1.e	. . . 4 ⊢ ∼ = (𝑅 ~RL 𝑆)
20		erlcl1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
21		erldi.1	. . . . 5 ⊢ 0 = (0g‘𝑅)
22		erldi.2	. . . . 5 ⊢ · = (.r‘𝑅)
23		erldi.3	. . . . 5 ⊢ − = (-g‘𝑅)
24		eqid 2726	. . . . 5 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
25		eqid 2726	. . . . 5 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )} = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )}
26		erlcl1.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
27	20, 21, 22, 23, 24, 25, 26	erlval 33118	. . . 4 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
28	19, 27	eqtrid 2778	. . 3 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
29		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → 𝑎 = 𝑈)
30	29	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = (1st ‘𝑈))
31	1	fveq2d 6897	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑈) = (1st ‘⟨𝐸, 𝐺⟩))
32		op1stg 8007	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
33	2, 3, 32	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘⟨𝐸, 𝐺⟩) = 𝐸)
34	31, 33	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑈) = 𝐸)
35	34	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑈) = 𝐸)
36	30, 35	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑎) = 𝐸)
37		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → 𝑏 = 𝑉)
38	37	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = (2nd ‘𝑉))
39	6	fveq2d 6897	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑉) = (2nd ‘⟨𝐹, 𝐻⟩))
40		op2ndg 8008	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
41	7, 8, 40	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘⟨𝐹, 𝐻⟩) = 𝐻)
42	39, 41	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑉) = 𝐻)
43	42	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑉) = 𝐻)
44	38, 43	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑏) = 𝐻)
45	36, 44	oveq12d 7434	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑎) · (2nd ‘𝑏)) = (𝐸 · 𝐻))
46	37	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = (1st ‘𝑉))
47	6	fveq2d 6897	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘𝑉) = (1st ‘⟨𝐹, 𝐻⟩))
48		op1stg 8007	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐻 ∈ 𝑆) → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
49	7, 8, 48	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (1st ‘⟨𝐹, 𝐻⟩) = 𝐹)
50	47, 49	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝜑 → (1st ‘𝑉) = 𝐹)
51	50	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑉) = 𝐹)
52	46, 51	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (1st ‘𝑏) = 𝐹)
53	29	fveq2d 6897	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = (2nd ‘𝑈))
54	1	fveq2d 6897	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘𝑈) = (2nd ‘⟨𝐸, 𝐺⟩))
55		op2ndg 8008	. . . . . . . . . . . 12 ⊢ ((𝐸 ∈ 𝐵 ∧ 𝐺 ∈ 𝑆) → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
56	2, 3, 55	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (2nd ‘⟨𝐸, 𝐺⟩) = 𝐺)
57	54, 56	eqtrd 2766	. . . . . . . . . 10 ⊢ (𝜑 → (2nd ‘𝑈) = 𝐺)
58	57	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑈) = 𝐺)
59	53, 58	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (2nd ‘𝑎) = 𝐺)
60	52, 59	oveq12d 7434	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((1st ‘𝑏) · (2nd ‘𝑎)) = (𝐹 · 𝐺))
61	45, 60	oveq12d 7434	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = ((𝐸 · 𝐻) − (𝐹 · 𝐺)))
62	61	oveq2d 7432	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))))
63	62	eqeq1d 2728	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
64	63	rexbidv 3169	. . 3 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 ))
65	28, 64	brab2d 32527	. 2 ⊢ (𝜑 → (𝑈 ∼ 𝑉 ↔ ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · ((𝐸 · 𝐻) − (𝐹 · 𝐺))) = 0 )))
66	18, 65	mpbird 256	1 ⊢ (𝜑 → 𝑈 ∼ 𝑉)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∃wrex 3060   ⊆ wss 3946  ⟨cop 4629   class class class wbr 5145  {copab 5207   × cxp 5672  ‘cfv 6546  (class class class)co 7416  1^st c1st 7993  2^nd c2nd 7994  Basecbs 17208  ._rcmulr 17262  0_gc0g 17449  -_gcsg 18925   ~_RL cerl 33113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-erl 33115

This theorem is referenced by:  erlbr2d  33124  erler  33125  rlocaddval  33128  rlocmulval  33129  rloccring  33130