reldmrloc - Mathbox for Thierry Arnoux

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Theorem reldmrloc 33338

Description: Ring localization is a proper operator, so it can be used with ovprc1 7395. (Contributed by Thierry Arnoux, 10-May-2025.)

**Assertion**
Ref	Expression
reldmrloc	⊢ Rel dom RLocal

Proof of Theorem reldmrloc Dummy variables 𝑎 𝑏 𝑘 𝑟 𝑠 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rloc 33337	. 2 ⊢ RLocal = (𝑟 ∈ V, 𝑠 ∈ V ↦ ⦋(._r‘𝑟) / 𝑥⦌⦋((Base‘𝑟) × 𝑠) / 𝑤⦌((({⟨(Base‘ndx), 𝑤⟩, ⟨(+_g‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨(((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(+_g‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎))), ((2^nd ‘𝑎)𝑥(2^nd ‘𝑏))⟩)⟩, ⟨(._r‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ ⟨((1^st ‘𝑎)𝑥(1^st ‘𝑏)), ((2^nd ‘𝑎)𝑥(2^nd ‘𝑏))⟩)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑟)⟩, ⟨( ·_𝑠 ‘ndx), (𝑘 ∈ (Base‘(Scalar‘𝑟)), 𝑎 ∈ 𝑤 ↦ ⟨(𝑘( ·_𝑠 ‘𝑟)(1^st ‘𝑎)), (2^nd ‘𝑎)⟩)⟩, ⟨(·_𝑖‘ndx), ∅⟩}) ∪ {⟨(TopSet‘ndx), ((TopSet‘𝑟) ×_t ((TopSet‘𝑟) ↾_t 𝑠))⟩, ⟨(le‘ndx), {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ 𝑤 ∧ 𝑏 ∈ 𝑤) ∧ ((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(le‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎)))}⟩, ⟨(dist‘ndx), (𝑎 ∈ 𝑤, 𝑏 ∈ 𝑤 ↦ (((1^st ‘𝑎)𝑥(2^nd ‘𝑏))(dist‘𝑟)((1^st ‘𝑏)𝑥(2^nd ‘𝑎))))⟩}) /_s (𝑟 ~_RL 𝑠)))
2	1	reldmmpo 7490	1 ⊢ Rel dom RLocal

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 ∈ wcel 2119 Vcvv 3431 ⦋csb 3831 ∪ cun 3881 ∅c0 4261 {ctp 4559 ⟨cop 4561 class class class wbr 5072 {copab 5134 × cxp 5616 dom cdm 5618 Rel wrel 5623 ‘cfv 6485 (class class class)co 7356 ∈ cmpo 7358 1^st c1st 7929 2^nd c2nd 7930 ndxcnx 17154 Basecbs 17170 +_gcplusg 17211 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 ·_𝑖cip 17216 TopSetcts 17217 lecple 17218 distcds 17220 ↾_t crest 17374 /_s cqus 17460 ×_t ctx 23543 ~_RL cerl 33334 RLocal crloc 33335

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-pr 5362

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-rab 3392 df-v 3433 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-br 5073 df-opab 5135 df-xp 5624 df-rel 5625 df-dm 5628 df-oprab 7360 df-mpo 7361 df-rloc 33337

This theorem is referenced by: fracval 33388

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