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Theorem fracval 33285
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.
StepHypRef Expression
1 df-frac 33284 . . 3 Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟)))
2 id 22 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
3 fveq2 6906 . . . . 5 (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅))
42, 3oveq12d 7448 . . . 4 (𝑟 = 𝑅 → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
54adantl 481 . . 3 ((𝑅 ∈ V ∧ 𝑟 = 𝑅) → (𝑟 RLocal (RLReg‘𝑟)) = (𝑅 RLocal (RLReg‘𝑅)))
6 id 22 . . 3 (𝑅 ∈ V → 𝑅 ∈ V)
7 ovexd 7465 . . 3 (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V)
81, 5, 6, 7fvmptd2 7023 . 2 (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
9 fvprc 6898 . . 3 𝑅 ∈ V → ( Frac ‘𝑅) = ∅)
10 reldmrloc 33243 . . . 4 Rel dom RLocal
1110ovprc1 7469 . . 3 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅)
129, 11eqtr4d 2777 . 2 𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)))
138, 12pm2.61i 182 1 ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  cfv 6562  (class class class)co 7430  RLRegcrlreg 20707   RLocal crloc 33240   Frac cfrac 33283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-rloc 33242  df-frac 33284
This theorem is referenced by:  fracbas  33286  fracf1  33288  fracfld  33289  zringfrac  33561
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