Theorem erld2 33595

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 2762	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
5	4, 1	mgpbas 20227	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
6	5	submss 18873	. . . 4 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
7	3, 6	syl 18	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8		eqid 2762	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		erld2.t	. . 3 ⊢ · = (.r‘𝑅)
10		eqid 2762	. . 3 ⊢ (-g‘𝑅) = (-g‘𝑅)
11		erld2.1	. . . 4 ⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )
12		eqid 2762	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
13		eqid 2762	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
14		erld2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
15	1, 8, 12, 9, 10, 13, 2, 14, 3	erler 33594	. . . . 5 ⊢ (𝜑 → ∼ Er (𝐵 × 𝑆))
16		erld2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		erld2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
18	16, 17	opelxpd 5699	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝑆))
19	15, 18	erth 8747	. . . 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 33591	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅))
22	14	crngringd 20334	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ringgrpd 20330	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
24	23	ad2antrr 738	. . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑅 ∈ Ring)
26	25	adantr 485	. . . . . 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 3936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝐵)
28	27	adantr 485	. . . . . 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 3937	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31	1, 9, 22, 16, 30	ringcld 20345	. . . . . . . 8 ⊢ (𝜑 → (𝑋 · 𝑊) ∈ 𝐵)
32	31	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑋 · 𝑊) ∈ 𝐵)
33	32	adantr 485	. . . . . 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 20345	. . . . 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 3937	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
37	1, 9, 22, 35, 36	ringcld 20345	. . . . . . . 8 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ 𝐵)
38	37	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑍 · 𝑌) ∈ 𝐵)
39	38	adantr 485	. . . . . 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 20345	. . . . 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 7996	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
42	16, 17, 41	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
43		op2ndg 7997	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
44	35, 29, 43	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
45	42, 44	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩)) = (𝑋 · 𝑊))
46		op1stg 7996	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
47	35, 29, 46	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
48		op2ndg 7997	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
49	16, 17, 48	syl2anc 595	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
50	47, 49	oveq12d 7430	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)) = (𝑍 · 𝑌))
51	45, 50	oveq12d 7430	. . . . . . . . . 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 7428	. . . . . . . . 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 485	. . . . . . . 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 20397	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))) = ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))))
55	53, 54	eqtrd 2797	. . . . . . 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 2764	. . . . . 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 481	. . . . 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 19098	. . . . . 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 481	. . . . 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 1399	. . . 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 417	. . 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 3177	. 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 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904  ⟨cop 4594   class class class wbr 5108   × cxp 5658  ‘cfv 6536  (class class class)co 7412  1^st c1st 7982  2^nd c2nd 7983  [cec 8690  Basecbs 17275  ._rcmulr 17317  0_gc0g 17498  SubMndcsubmnd 18846  Grpcgrp 19006  -_gcsg 19008  mulGrpcmgp 20222  1_rcur 20269  Ringcrg 20321  CRingccrg 20322   ~_RL cerl 33582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-ec 8694  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-0g 17500  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-cring 20324  df-erl 33584

This theorem is used by:  rlocisunit  33605