reldmrloc - Mathbox for Thierry Arnoux

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Theorem reldmrloc 33288

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 33287	. 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 395 ∈ wcel 2113 Vcvv 3438 ⦋csb 3847 ∪ cun 3897 ∅c0 4283 {ctp 4582 ⟨cop 4584 class class class wbr 5096 {copab 5158 × cxp 5620 dom cdm 5622 Rel wrel 5627 ‘cfv 6490 (class class class)co 7356 ∈ cmpo 7358 1^st c1st 7929 2^nd c2nd 7930 ndxcnx 17118 Basecbs 17134 +_gcplusg 17175 ._rcmulr 17176 Scalarcsca 17178 ·_𝑠 cvsca 17179 ·_𝑖cip 17180 TopSetcts 17181 lecple 17182 distcds 17184 ↾_t crest 17338 /_s cqus 17424 ×_t ctx 23502 ~_RL cerl 33284 RLocal crloc 33285

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-sep 5239 ax-nul 5249 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-rab 3398 df-v 3440 df-dif 3902 df-un 3904 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-br 5097 df-opab 5159 df-xp 5628 df-rel 5629 df-dm 5632 df-oprab 7360 df-mpo 7361 df-rloc 33287

This theorem is referenced by: fracval 33335

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