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Mirrors > Home > MPE Home > Th. List > qnumval | Structured version Visualization version 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 2728 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | |
2 | 1 | anbi2d 628 | . . . 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 6885 | . 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 16670 | . 2 ⊢ numer = (𝑎 ∈ ℚ ↦ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | |
6 | fvex 6894 | . 2 ⊢ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6988 | 1 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩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 1533 ∈ wcel 2098 × cxp 5664 ‘cfv 6533 ℩crio 7356 (class class class)co 7401 1st c1st 7966 2nd c2nd 7967 1c1 11107 / cdiv 11868 ℕcn 12209 ℤcz 12555 ℚcq 12929 gcd cgcd 16432 numercnumer 16668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-riota 7357 df-numer 16670 |
This theorem is referenced by: qnumdencl 16674 fnum 16677 qnumdenbi 16679 |
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