| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fracval | Structured version Visualization version GIF version | ||
| Description: Value of the field of fractions. (Contributed by Thierry Arnoux, 5-May-2025.) |
| Ref | Expression |
|---|---|
| fracval | ⊢ ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅)) |
| 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)) |
| Colors of variables: wff setvar class |
| 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 |
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