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| Mirrors > Home > MPE Home > Th. List > qdenval | Structured version Visualization version GIF version | ||
| Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
| Ref | Expression |
|---|---|
| qdenval | ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq1 2738 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | |
| 2 | 1 | anbi2d 630 | . . . 4 ⊢ (𝑎 = 𝐴 → ((((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥))) ↔ (((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) |
| 3 | 2 | riotabidv 7359 | . . 3 ⊢ (𝑎 = 𝐴 → (℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))) = (℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) |
| 4 | 3 | fveq2d 6877 | . 2 ⊢ (𝑎 = 𝐴 → (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥))))) = (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
| 5 | df-denom 16742 | . 2 ⊢ denom = (𝑎 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | |
| 6 | fvex 6886 | . 2 ⊢ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) ∈ V | |
| 7 | 4, 5, 6 | fvmpt 6983 | 1 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 × cxp 5650 ‘cfv 6528 ℩crio 7356 (class class class)co 7400 1st c1st 7981 2nd c2nd 7982 1c1 11123 / cdiv 11887 ℕcn 12233 ℤcz 12581 ℚcq 12957 gcd cgcd 16500 denomcdenom 16740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5264 ax-nul 5274 ax-pr 5400 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3414 df-v 3459 df-dif 3927 df-un 3929 df-ss 3941 df-nul 4307 df-if 4499 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4882 df-br 5118 df-opab 5180 df-mpt 5200 df-id 5546 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-iota 6481 df-fun 6530 df-fv 6536 df-riota 7357 df-denom 16742 |
| This theorem is referenced by: qnumdencl 16745 fden 16749 qnumdenbi 16750 |
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