reldmrloc - Mathbox for Thierry Arnoux

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Theorem reldmrloc 33047

Description: Ring localization is a proper operator, so it can be used with ovprc1 7458. (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 33046	. 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 7555	1 ⊢ Rel dom RLocal

Colors of variables: wff setvar class

Syntax hints: ∧ wa 394 ∈ wcel 2098 Vcvv 3461 ⦋csb 3889 ∪ cun 3942 ∅c0 4322 {ctp 4634 ⟨cop 4636 class class class wbr 5149 {copab 5211 × cxp 5676 dom cdm 5678 Rel wrel 5683 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 1^st c1st 7992 2^nd c2nd 7993 ndxcnx 17165 Basecbs 17183 +_gcplusg 17236 ._rcmulr 17237 Scalarcsca 17239 ·_𝑠 cvsca 17240 ·_𝑖cip 17241 TopSetcts 17242 lecple 17243 distcds 17245 ↾_t crest 17405 /_s cqus 17490 ×_t ctx 23508 ~_RL cerl 33043 RLocal crloc 33044

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-dm 5688 df-oprab 7423 df-mpo 7424 df-rloc 33046

This theorem is referenced by: fracval 33090

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