Theorem erld2 33401

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 2756	. . . . . 6 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
5	4, 1	mgpbas 20167	. . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅))
6	5	submss 18819	. . . 4 ⊢ (𝑆 ∈ (SubMnd‘(mulGrp‘𝑅)) → 𝑆 ⊆ 𝐵)
7	3, 6	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8		eqid 2756	. . 3 ⊢ (0g‘𝑅) = (0g‘𝑅)
9		erld2.t	. . 3 ⊢ · = (.r‘𝑅)
10		eqid 2756	. . 3 ⊢ (-g‘𝑅) = (-g‘𝑅)
11		erld2.1	. . . 4 ⊢ (𝜑 → [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ )
12		eqid 2756	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
13		eqid 2756	. . . . . 6 ⊢ (𝐵 × 𝑆) = (𝐵 × 𝑆)
14		erld2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CRing)
15	1, 8, 12, 9, 10, 13, 2, 14, 3	erler 33400	. . . . 5 ⊢ (𝜑 → ∼ Er (𝐵 × 𝑆))
16		erld2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		erld2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
18	16, 17	opelxpd 5679	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝑆))
19	15, 18	erth 8721	. . . 4 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∼ ⟨𝑍, 𝑊⟩ ↔ [⟨𝑋, 𝑌⟩] ∼ = [⟨𝑍, 𝑊⟩] ∼ ))
20	11, 19	mpbird 259	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∼ ⟨𝑍, 𝑊⟩)
21	1, 2, 7, 8, 9, 10, 20	erldi 33397	. 2 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩))(-g‘𝑅)((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)))) = (0g‘𝑅))
22	14	crngringd 20268	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
23	22	ringgrpd 20264	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
24	23	ad2antrr 734	. . . . 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 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑅 ∈ Ring)
26	25	adantr 483	. . . . . 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 3931	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → 𝑡 ∈ 𝐵)
28	27	adantr 483	. . . . . 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 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
31	1, 9, 22, 16, 30	ringcld 20282	. . . . . . . 8 ⊢ (𝜑 → (𝑋 · 𝑊) ∈ 𝐵)
32	31	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑋 · 𝑊) ∈ 𝐵)
33	32	adantr 483	. . . . . 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 20282	. . . . 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 3932	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
37	1, 9, 22, 35, 36	ringcld 20282	. . . . . . . 8 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ 𝐵)
38	37	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑍 · 𝑌) ∈ 𝐵)
39	38	adantr 483	. . . . . 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 20282	. . . . 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 7971	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
42	16, 17, 41	syl2anc 592	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
43		op2ndg 7972	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
44	35, 29, 43	syl2anc 592	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
45	42, 44	oveq12d 7403	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑋, 𝑌⟩) · (2nd ‘⟨𝑍, 𝑊⟩)) = (𝑋 · 𝑊))
46		op1stg 7971	. . . . . . . . . . . . 13 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝑆) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
47	35, 29, 46	syl2anc 592	. . . . . . . . . . . 12 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
48		op2ndg 7972	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑆) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
49	16, 17, 48	syl2anc 592	. . . . . . . . . . . 12 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
50	47, 49	oveq12d 7403	. . . . . . . . . . 11 ⊢ (𝜑 → ((1st ‘⟨𝑍, 𝑊⟩) · (2nd ‘⟨𝑋, 𝑌⟩)) = (𝑍 · 𝑌))
51	45, 50	oveq12d 7403	. . . . . . . . . 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 7401	. . . . . . . . 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 483	. . . . . . . 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 20329	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑆) → (𝑡 · ((𝑋 · 𝑊)(-g‘𝑅)(𝑍 · 𝑌))) = ((𝑡 · (𝑋 · 𝑊))(-g‘𝑅)(𝑡 · (𝑍 · 𝑌))))
55	53, 54	eqtrd 2791	. . . . . . 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 2758	. . . . . 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 479	. . . . 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 19044	. . . . . 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 479	. . . . 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 1388	. . . 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 415	. . 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 3169	. 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 15	1 ⊢ (𝜑 → ∃𝑡 ∈ 𝑆 (𝑡 · (𝑋 · 𝑊)) = (𝑡 · (𝑍 · 𝑌)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∃wrex 3080   ⊆ wss 3899  ⟨cop 4582   class class class wbr 5094   × cxp 5638  ‘cfv 6510  (class class class)co 7385  1^st c1st 7957  2^nd c2nd 7958  [cec 8664  Basecbs 17221  ._rcmulr 17263  0_gc0g 17444  SubMndcsubmnd 18792  Grpcgrp 18951  -_gcsg 18953  mulGrpcmgp 20162  1_rcur 20203  Ringcrg 20255  CRingccrg 20256   ~_RL cerl 33388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-cnex 11119  ax-resscn 11120  ax-1cn 11121  ax-icn 11122  ax-addcl 11123  ax-addrcl 11124  ax-mulcl 11125  ax-mulrcl 11126  ax-mulcom 11127  ax-addass 11128  ax-mulass 11129  ax-distr 11130  ax-i2m1 11131  ax-1ne0 11132  ax-1rid 11133  ax-rnegex 11134  ax-rrecex 11135  ax-cnre 11136  ax-pre-lttri 11137  ax-pre-lttrn 11138  ax-pre-ltadd 11139  ax-pre-mulgt0 11140

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-nel 3056  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-er 8666  df-ec 8668  df-en 8917  df-dom 8918  df-sdom 8919  df-pnf 11208  df-mnf 11209  df-xr 11210  df-ltxr 11211  df-le 11212  df-sub 11406  df-neg 11407  df-nn 12201  df-2 12270  df-sets 17176  df-slot 17194  df-ndx 17206  df-base 17222  df-ress 17243  df-plusg 17275  df-0g 17446  df-mgm 18650  df-sgrp 18729  df-mnd 18745  df-submnd 18794  df-grp 18954  df-minusg 18955  df-sbg 18956  df-cmn 19798  df-abl 19799  df-mgp 20163  df-rng 20175  df-ur 20204  df-ring 20257  df-cring 20258  df-erl 33390

This theorem is referenced by:  rlocisunit  33411