Theorem fracval 33916

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 33915	. . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2		id 23	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6890	. . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
4	2, 3	oveq12d 7442	. . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
5	4	adantl 487	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑟 = 𝑅) → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
6		id 23	. . 3 ⊢ (𝑅 ∈ V → 𝑅 ∈ V)
7		ovexd 7459	. . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
8	1, 5, 6, 7	fvmptd2 7007	. 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9		fvprc 6882	. . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10		reldmrloc 33868	. . . 4 ⊢ Rel dom RLocal
11	10	ovprc1 7463	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
12	9, 11	eqtr4d 2798	. 2 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
13	8, 12	pm2.61i 184	1 ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4277  ‘cfv 6543  (class class class)co 7424  RLRegcrlreg 20952   RLocal crloc 33865   Frac cfrac 33914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6497  df-fun 6545  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-rloc 33867  df-frac 33915

This theorem is used by:  fracbas  33917  fracf1  33919  fracfld  33920  zringfrac  34136