Theorem fracval 33306

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 33305	. . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6906	. . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
4	2, 3	oveq12d 7449	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
5	4	adantl 481	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑟 = 𝑅) → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
6		id 22	. . 3 ⊢ (𝑅 ∈ V → 𝑅 ∈ V)
7		ovexd 7466	. . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
8	1, 5, 6, 7	fvmptd2 7024	. 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9		fvprc 6898	. . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10		reldmrloc 33261	. . . 4 ⊢ Rel dom RLocal
11	10	ovprc1 7470	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
12	9, 11	eqtr4d 2780	. 2 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
13	8, 12	pm2.61i 182	1 ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  ‘cfv 6561  (class class class)co 7431  RLRegcrlreg 20691   RLocal crloc 33258   Frac cfrac 33304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rloc 33260  df-frac 33305

This theorem is referenced by:  fracbas  33307  fracf1  33309  fracfld  33310  zringfrac  33582