Theorem fracval 33567

Description: Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.)

Assertion

Ref	Expression
fracval	⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))

Proof of Theorem fracval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frac 33566	. . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2		id 23	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6882	. . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
4	2, 3	oveq12d 7429	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
5	4	adantl 486	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑟 = 𝑅) → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
6		id 23	. . 3 ⊢ (𝑅 ∈ V → 𝑅 ∈ V)
7		ovexd 7446	. . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
8	1, 5, 6, 7	fvmptd2 6999	. 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9		fvprc 6874	. . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10		reldmrloc 33517	. . . 4 ⊢ Rel dom RLocal
11	10	ovprc1 7450	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
12	9, 11	eqtr4d 2807	. 2 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
13	8, 12	pm2.61i 184	1 ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  ‘cfv 6537  (class class class)co 7411  RLRegcrlreg 20775   RLocal crloc 33514   Frac cfrac 33565

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-nul 5271  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-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  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-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rloc 33516  df-frac 33566

This theorem is referenced by:  fracbas  33568  fracf1  33570  fracfld  33571  zringfrac  33788