Theorem fracval 33383

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 33382	. . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2		id 22	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
3		fveq2 6835	. . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
4	2, 3	oveq12d 7379	. . . 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 7396	. . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
8	1, 5, 6, 7	fvmptd2 6951	. 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9		fvprc 6827	. . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10		reldmrloc 33336	. . . 4 ⊢ Rel dom RLocal
11	10	ovprc1 7400	. . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
12	9, 11	eqtr4d 2775	. 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 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ‘cfv 6493  (class class class)co 7361  RLRegcrlreg 20662   RLocal crloc 33333   Frac cfrac 33381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-rloc 33335  df-frac 33382

This theorem is referenced by:  fracbas  33384  fracf1  33386  fracfld  33387  zringfrac  33632