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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 2177 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | |
2 | 1 | anbi2d 461 | . . . 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 5811 | . . 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 5500 | . 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 12137 | . 2 ⊢ numer = (𝑎 ∈ ℚ ↦ (1st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | |
6 | zex 9221 | . . . 4 ⊢ ℤ ∈ V | |
7 | nnex 8884 | . . . 4 ⊢ ℕ ∈ V | |
8 | 6, 7 | xpex 4726 | . . 3 ⊢ (ℤ × ℕ) ∈ V |
9 | riotaexg 5813 | . . 3 ⊢ ((ℤ × ℕ) ∈ V → (℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) ∈ V) | |
10 | 1stexg 6146 | . . 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 5573 | 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 1348 ∈ wcel 2141 Vcvv 2730 × cxp 4609 ‘cfv 5198 ℩crio 5808 (class class class)co 5853 1st c1st 6117 2nd c2nd 6118 1c1 7775 / cdiv 8589 ℕcn 8878 ℤcz 9212 ℚcq 9578 gcd cgcd 11897 numercnumer 12135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866 ax-1re 7868 ax-addrcl 7871 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 df-riota 5809 df-ov 5856 df-1st 6119 df-neg 8093 df-inn 8879 df-z 9213 df-numer 12137 |
This theorem is referenced by: qnumdencl 12141 fnum 12144 qnumdenbi 12146 |
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