Theorem fracval 33262

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 33261	. . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6817	. . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
4	2, 3	oveq12d 7359	. . . 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 7376	. . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
8	1, 5, 6, 7	fvmptd2 6932	. 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9		fvprc 6809	. . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10		reldmrloc 33216	. . . 4 ⊢ Rel dom RLocal
11	10	ovprc1 7380	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
12	9, 11	eqtr4d 2769	. 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 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4278  ‘cfv 6476  (class class class)co 7341  RLRegcrlreg 20601   RLocal crloc 33213   Frac cfrac 33260

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 5229  ax-nul 5239  ax-pr 5365

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-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-rloc 33215  df-frac 33261

This theorem is referenced by:  fracbas  33263  fracf1  33265  fracfld  33266  zringfrac  33511