reldmrloc - Mathbox for Thierry Arnoux

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Theorem reldmrloc 33224

Description: Ring localization is a proper operator, so it can be used with ovprc1 7385. (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 33223	. 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 7480	1 ⊢ Rel dom RLocal

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∈ wcel 2111 Vcvv 3436 ⦋csb 3845 ∪ cun 3895 ∅c0 4280 {ctp 4577 ⟨cop 4579 class class class wbr 5089 {copab 5151 × cxp 5612 dom cdm 5614 Rel wrel 5619 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ndxcnx 17104 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 ·_𝑖cip 17166 TopSetcts 17167 lecple 17168 distcds 17170 ↾_t crest 17324 /_s cqus 17409 ×_t ctx 23475 ~_RL cerl 33220 RLocal crloc 33221

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-br 5090 df-opab 5152 df-xp 5620 df-rel 5621 df-dm 5624 df-oprab 7350 df-mpo 7351 df-rloc 33223

This theorem is referenced by: fracval 33270

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