Theorem erld2 33897

Description: Main property of the ring localization equivalence relation. (Contributed by Thierry Arnoux, 4-May-2025.)

Hypotheses

Ref	Expression
erld2.b	⊢ 𝐵 = (Base‘𝑅)
erld2.e	⊢ ∼ = (𝑅 ~RL 𝑆)
erld2.t	⊢ · = (.r‘𝑅)
erld2.r	⊢ (𝜑 → 𝑅 ∈ CRing)
erld2.s	⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
erld2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
erld2.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)
erld2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
erld2.w	⊢ (𝜑 → 𝑊 ∈ 𝑆)
erld2.1	⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )

Assertion

Ref	Expression
erld2	⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))

Distinct variable groups:   𝑡, ·   𝑡,𝐵   𝑡,𝑅   𝑡,𝑆   𝑡,𝑊   𝑡,𝑋   𝑡,𝑌   𝑡,𝑍   𝜑,𝑡

Allowed substitution hint:   ∼ (𝑡)

Proof of Theorem erld2

Step	Hyp	Ref	Expression
1		erld2.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		erld2.e	. . 3 ⊢ ∼ = (𝑅 ~RL 𝑆)
3		erld2.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)))
4		eqid 2763	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
5	4, 1	mgpbas 20395	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
6	5	submss 19033	. . . 4 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
7	3, 6	syl 18	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8		eqid 2763	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		erld2.t	. . 3 ⊢ · = (.r‘𝑅)
10		eqid 2763	. . 3 ⊢ (-g‘𝑅) = (-g‘𝑅)
11		erld2.1	. . . 4 ⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )
12		eqid 2763	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
13		eqid 2763	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
14		erld2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
15	1, 8, 12, 9, 10, 13, 2, 14, 3	erler 33896	. . . . 5 ⊢ (𝜑 → ∼ Er (𝐵 × 𝑆))
16		erld2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		erld2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
18	16, 17	opelxpd 5694	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝑆))
19	15, 18	erth 8800	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∼ ⟨𝑍, 𝑊⟩ ↔ [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ ))
20	11, 19	mpbird 260	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∼ ⟨𝑍, 𝑊⟩)
21	1, 2, 7, 8, 9, 10, 20	erldi 33893	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅))
22	14	crngringd 20503	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ringgrpd 20499	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
24	23	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → 𝑅 ∈ Grp)
25	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑅 ∈ Ring)
26	25	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → 𝑅 ∈ Ring)
27	7	sselda 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝐵)
28	27	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → 𝑡 ∈ 𝐵)
29		erld2.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝑆)
30	7, 29	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31	1, 9, 22, 16, 30	ringcld 20514	. . . . . . . 8 ⊢ (𝜑 → (𝑋 · 𝑊) ∈ 𝐵)
32	31	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑋 · 𝑊) ∈ 𝐵)
33	32	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → (𝑋 · 𝑊) ∈ 𝐵)
34	1, 9, 26, 28, 33	ringcld 20514	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → (𝑡 · (𝑋 · 𝑊)) ∈ 𝐵)
35		erld2.z	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
36	7, 17	sseldd 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
37	1, 9, 22, 35, 36	ringcld 20514	. . . . . . . 8 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ 𝐵)
38	37	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑍 · 𝑌) ∈ 𝐵)
39	38	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → (𝑍 · 𝑌) ∈ 𝐵)
40	1, 9, 26, 28, 39	ringcld 20514	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → (𝑡 · (𝑍 · 𝑌)) ∈ 𝐵)
41		op1stg 8037	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
42	16, 17, 41	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
43		op2ndg 8038	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
44	35, 29, 43	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
45	42, 44	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩)) = (𝑋 · 𝑊))
46		op1stg 8037	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
47	35, 29, 46	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
48		op2ndg 8038	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
49	16, 17, 48	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
50	47, 49	oveq12d 7461	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)) = (𝑍 · 𝑌))
51	45, 50	oveq12d 7461	. . . . . . . . . 10 ⊢ (𝜑 → (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩))) = ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌)))
52	51	oveq2d 7459	. . . . . . . . 9 ⊢ (𝜑 → (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))))
53	52	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))))
54	1, 9, 10, 25, 27, 32, 38	ringsubdi 20568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))) = ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))))
55	53, 54	eqtrd 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))))
56	55	eqeq1d 2765	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → ((𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅) ↔ ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))) = (0g‘𝑅)))
57	56	biimpa 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))) = (0g‘𝑅))
58	1, 8, 10	grpsubeq0 19265	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ (𝑡 · (𝑋 · 𝑊)) ∈ 𝐵 ∧ (𝑡 · (𝑍 · 𝑌)) ∈ 𝐵) → (((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))) = (0g‘𝑅) ↔ (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌))))
59	58	biimpa 482	. . . . 5 ⊢ (((𝑅 ∈ Grp ∧ (𝑡 · (𝑋 · 𝑊)) ∈ 𝐵 ∧ (𝑡 · (𝑍 · 𝑌)) ∈ 𝐵) ∧ ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))) = (0g‘𝑅)) → (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))
60	24, 34, 40, 57, 59	syl31anc 1401	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑆) ∧ (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅)) → (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))
61	60	ex 418	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → ((𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅) → (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌))))
62	61	reximdva 3179	. 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅) → ∃𝑡 ∈ 𝑆 (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌))))
63	21, 62	mpd 16	1 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  ⟨cop 4838   class class class wbr 5106   × cxp 5653  ‘cfv 6558  (class class class)co 7443  1^st c1st 8023  2^nd c2nd 8024  [cec 8743  Basecbs 17416  ._rcmulr 17458  0_gc0g 17639  SubMndcsubmnd 19006  Grpcgrp 19173  -_gcsg 19175  mulGrpcmgp 20390  1_rcur 20437  Ringcrg 20489  CRingccrg 20490   ~_RL cerl 33884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-er 8745  df-ec 8747  df-en 9002  df-dom 9003  df-sdom 9004  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-sets 17371  df-slot 17389  df-ndx 17401  df-base 17417  df-ress 17438  df-plusg 17470  df-0g 17641  df-mgm 18845  df-sgrp 18937  df-mnd 18953  df-submnd 19008  df-grp 19176  df-minusg 19177  df-sbg 19178  df-cmn 20026  df-abl 20027  df-mgp 20391  df-rng 20405  df-ur 20438  df-ring 20491  df-cring 20492  df-erl 33886

This theorem is used by:  rlocisunit  33907