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Mirrors > Home > ILE Home > Th. List > qnumval | GIF version |
Description: Value of the canonical numerator function. (Contributed by Stefan O'Rear, 13-Sep-2014.) |
Ref | Expression |
---|---|
qnumval | ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2119 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | |
2 | 1 | anbi2d 457 | . . . 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 5684 | . . 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 5377 | . 2 ⊢ (𝑎 = 𝐴 → (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥))))) = (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
5 | df-numer 11700 | . 2 ⊢ numer = (𝑎 ∈ ℚ ↦ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | |
6 | zex 8961 | . . . 4 ⊢ ℤ ∈ V | |
7 | nnex 8630 | . . . 4 ⊢ ℕ ∈ V | |
8 | 6, 7 | xpex 4612 | . . 3 ⊢ (ℤ × ℕ) ∈ V |
9 | riotaexg 5686 | . . 3 ⊢ ((ℤ × ℕ) ∈ V → (℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) ∈ V) | |
10 | 1stexg 6017 | . . 3 ⊢ ((℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) ∈ V → (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) ∈ V) | |
11 | 8, 9, 10 | mp2b 8 | . 2 ⊢ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) ∈ V |
12 | 4, 5, 11 | fvmpt 5450 | 1 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 Vcvv 2655 × cxp 4495 ‘cfv 5079 ℩crio 5681 (class class class)co 5726 1st c1st 5988 2nd c2nd 5989 1c1 7542 / cdiv 8339 ℕcn 8624 ℤcz 8952 ℚcq 9307 gcd cgcd 11477 numercnumer 11698 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-cnex 7630 ax-resscn 7631 ax-1re 7633 ax-addrcl 7636 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-sbc 2877 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fo 5085 df-fv 5087 df-riota 5682 df-ov 5729 df-1st 5990 df-neg 7853 df-inn 8625 df-z 8953 df-numer 11700 |
This theorem is referenced by: qnumdencl 11704 fnum 11707 qnumdenbi 11709 |
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