![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 33284 | . . 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 7448 | . . . 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 7465 | . . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V) | |
8 | 1, 5, 6, 7 | fvmptd2 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 | |
11 | 10 | ovprc1 7469 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅) |
12 | 9, 11 | eqtr4d 2777 | . 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 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 |
Copyright terms: Public domain | W3C validator |