Theorem erld2 33870

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 2760	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
5	4, 1	mgpbas 20366	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
6	5	submss 19005	. . . 4 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
7	3, 6	syl 18	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8		eqid 2760	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		erld2.t	. . 3 ⊢ · = (.r‘𝑅)
10		eqid 2760	. . 3 ⊢ (-g‘𝑅) = (-g‘𝑅)
11		erld2.1	. . . 4 ⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )
12		eqid 2760	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
13		eqid 2760	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
14		erld2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
15	1, 8, 12, 9, 10, 13, 2, 14, 3	erler 33869	. . . . 5 ⊢ (𝜑 → ∼ Er (𝐵 × 𝑆))
16		erld2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		erld2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
18	16, 17	opelxpd 5689	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝑆))
19	15, 18	erth 8773	. . . 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 33866	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅))
22	14	crngringd 20474	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ringgrpd 20470	. . . . . 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 3908	. . . . . . 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 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31	1, 9, 22, 16, 30	ringcld 20485	. . . . . . . 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 20485	. . . . 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 3909	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
37	1, 9, 22, 35, 36	ringcld 20485	. . . . . . . 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 20485	. . . . 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 8013	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
42	16, 17, 41	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
43		op2ndg 8014	. . . . . . . . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩)) = (𝑋 · 𝑊))
46		op1stg 8013	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
47	35, 29, 46	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
48		op2ndg 8014	. . . . . . . . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)) = (𝑍 · 𝑌))
51	45, 50	oveq12d 7438	. . . . . . . . . 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 7436	. . . . . . . . 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 20539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))) = ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))))
55	53, 54	eqtrd 2795	. . . . . . 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 2762	. . . . . 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 19237	. . . . . 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 1400	. . . 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 3175	. 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 1569   ∈ wcel 2144  ∃wrex 3086   ⊆ wss 3895  ⟨cop 4589   class class class wbr 5102   × cxp 5648  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  [cec 8716  Basecbs 17388  ._rcmulr 17430  0_gc0g 17611  SubMndcsubmnd 18978  Grpcgrp 19145  -_gcsg 19147  mulGrpcmgp 20361  1_rcur 20408  Ringcrg 20460  CRingccrg 20461   ~_RL cerl 33857

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11257  ax-resscn 11258  ax-1cn 11259  ax-icn 11260  ax-addcl 11261  ax-addrcl 11262  ax-mulcl 11263  ax-mulrcl 11264  ax-mulcom 11265  ax-addass 11266  ax-mulass 11267  ax-distr 11268  ax-i2m1 11269  ax-1ne0 11270  ax-1rid 11271  ax-rnegex 11272  ax-rrecex 11273  ax-cnre 11274  ax-pre-lttri 11275  ax-pre-lttrn 11276  ax-pre-ltadd 11277  ax-pre-mulgt0 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-er 8718  df-ec 8720  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11346  df-mnf 11347  df-xr 11348  df-ltxr 11349  df-le 11350  df-sub 11544  df-neg 11545  df-nn 12337  df-2 12406  df-sets 17343  df-slot 17361  df-ndx 17373  df-base 17389  df-ress 17410  df-plusg 17442  df-0g 17613  df-mgm 18817  df-sgrp 18909  df-mnd 18925  df-submnd 18980  df-grp 19148  df-minusg 19149  df-sbg 19150  df-cmn 19997  df-abl 19998  df-mgp 20362  df-rng 20376  df-ur 20409  df-ring 20462  df-cring 20463  df-erl 33859

This theorem is used by:  rlocisunit  33880