| 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 33385 | . . 3 ⊢ Frac = (𝑟 ∈ V ↦ (𝑟 RLocal (RLReg‘𝑟))) | |
| 2 | id 22 | . . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
| 3 | fveq2 6834 | . . . . 5 ⊢ (𝑟 = 𝑅 → (RLReg‘𝑟) = (RLReg‘𝑅)) | |
| 4 | 2, 3 | oveq12d 7376 | . . . 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 7393 | . . 3 ⊢ (𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) ∈ V) | |
| 8 | 1, 5, 6, 7 | fvmptd2 6949 | . 2 ⊢ (𝑅 ∈ V → ( Frac ‘𝑅) = (𝑅 RLocal (RLReg‘𝑅))) |
| 9 | fvprc 6826 | . . 3 ⊢ (¬ 𝑅 ∈ V → ( Frac ‘𝑅) = ∅) | |
| 10 | reldmrloc 33339 | . . . 4 ⊢ Rel dom RLocal | |
| 11 | 10 | ovprc1 7397 | . . 3 ⊢ (¬ 𝑅 ∈ V → (𝑅 RLocal (RLReg‘𝑅)) = ∅) |
| 12 | 9, 11 | eqtr4d 2774 | . 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 1541 ∈ wcel 2113 Vcvv 3440 ∅c0 4285 ‘cfv 6492 (class class class)co 7358 RLRegcrlreg 20624 RLocal crloc 33336 Frac cfrac 33384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-rloc 33338 df-frac 33385 |
| This theorem is referenced by: fracbas 33387 fracf1 33389 fracfld 33390 zringfrac 33635 |
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