Theorem erldi 33219

Description: Main property of the ring localization equivalence 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)
erldi.4	⊢ (𝜑 → 𝑈 ∼ 𝑉)

Assertion

Ref	Expression
erldi	⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 )

Distinct variable groups:   𝑡, ·   𝑡,𝐵   𝑡,𝑅   𝑡,𝑆   𝑡,𝑈   𝑡,𝑉

Allowed substitution hints:   𝜑(𝑡)   ∼ (𝑡)   − (𝑡)   0 (𝑡)

Proof of Theorem erldi

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

Step	Hyp	Ref	Expression
1		erldi.4	. . 3 ⊢ (𝜑 → 𝑈 ∼ 𝑉)
2		erlcl1.e	. . . . 5 ⊢ ∼ = (𝑅 ~RL 𝑆)
3		erlcl1.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4		erldi.1	. . . . . 6 ⊢ 0 = (0g‘𝑅)
5		erldi.2	. . . . . 6 ⊢ · = (.r‘𝑅)
6		erldi.3	. . . . . 6 ⊢ − = (-g‘𝑅)
7		eqid 2730	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
8		eqid 2730	. . . . . 6 ⊢ {⟨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 )}
9		erlcl1.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
10	3, 4, 5, 6, 7, 8, 9	erlval 33215	. . . . 5 ⊢ (𝜑 → (𝑅 ~RL 𝑆) = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
11	2, 10	eqtrid 2777	. . . 4 ⊢ (𝜑 → ∼ = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝐵 × 𝑆) ∧ 𝑏 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 )})
12		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑎 = 𝑈)
13	12	fveq2d 6821	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑎) = (1st ‘𝑈))
14		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → 𝑏 = 𝑉)
15	14	fveq2d 6821	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑏) = (2nd ‘𝑉))
16	13, 15	oveq12d 7359	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑎) · (2nd ‘𝑏)) = ((1st ‘𝑈) · (2nd ‘𝑉)))
17	14	fveq2d 6821	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (1st ‘𝑏) = (1st ‘𝑉))
18	12	fveq2d 6821	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (2nd ‘𝑎) = (2nd ‘𝑈))
19	17, 18	oveq12d 7359	. . . . . . . . 9 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((1st ‘𝑏) · (2nd ‘𝑎)) = ((1st ‘𝑉) · (2nd ‘𝑈)))
20	16, 19	oveq12d 7359	. . . . . . . 8 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎))) = (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈))))
21	20	oveq2d 7357	. . . . . . 7 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))))
22	21	eqeq1d 2732	. . . . . 6 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → ((𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 ))
23	22	rexbidv 3154	. . . . 5 ⊢ ((𝑎 = 𝑈 ∧ 𝑏 = 𝑉) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 ))
24	23	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑎 = 𝑈 ∧ 𝑏 = 𝑉)) → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑎) · (2nd ‘𝑏)) − ((1st ‘𝑏) · (2nd ‘𝑎)))) = 0 ↔ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 ))
25	11, 24	brab2d 32578	. . 3 ⊢ (𝜑 → (𝑈 ∼ 𝑉 ↔ ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 )))
26	1, 25	mpbid 232	. 2 ⊢ (𝜑 → ((𝑈 ∈ (𝐵 × 𝑆) ∧ 𝑉 ∈ (𝐵 × 𝑆)) ∧ ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 ))
27	26	simprd 495	1 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘𝑈) · (2nd ‘𝑉)) − ((1st ‘𝑉) · (2nd ‘𝑈)))) = 0 )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃wrex 3054   ⊆ wss 3900   class class class wbr 5089  {copab 5151   × cxp 5612  ‘cfv 6477  (class class class)co 7341  1^st c1st 7914  2^nd c2nd 7915  Basecbs 17112  ._rcmulr 17154  0_gc0g 17335  -_gcsg 18840   ~_RL cerl 33210

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-erl 33212

This theorem is referenced by:  rlocaddval  33225  rlocmulval  33226  rlocf1  33230  fracfld  33264