reldmrloc - Mathbox for Thierry Arnoux

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Theorem reldmrloc 33518

Description: Ring localization is a proper operator, so it can be used with ovprc1 7450. (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 33517	. 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 7545	1 ⊢ Rel dom RLocal

Colors of variables: wff setvar class

Syntax hints: ∧ wa 400 ∈ wcel 2149 Vcvv 3463 ⦋csb 3861 ∪ cun 3911 ∅c0 4294 {ctp 4598 ⟨cop 4600 class class class wbr 5113 {copab 5177 × cxp 5660 dom cdm 5662 Rel wrel 5667 ‘cfv 6537 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7984 2^nd c2nd 7985 ndxcnx 17253 Basecbs 17269 +_gcplusg 17310 ._rcmulr 17311 Scalarcsca 17313 ·_𝑠 cvsca 17314 ·_𝑖cip 17315 TopSetcts 17316 lecple 17317 distcds 17319 ↾_t crest 17473 /_s cqus 17559 ×_t ctx 23686 ~_RL cerl 33514 RLocal crloc 33515

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-pr 5405

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-rab 3424 df-v 3465 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-sn 4595 df-pr 4597 df-op 4601 df-br 5114 df-opab 5178 df-xp 5668 df-rel 5669 df-dm 5672 df-oprab 7415 df-mpo 7416 df-rloc 33517

This theorem is referenced by: fracval 33568

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