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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 2735 | . . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)) ↔ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥)))) | |
2 | 1 | anbi2d 629 | . . . 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 7351 | . . 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 6882 | . 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 16654 | . 2 ⊢ denom = (𝑎 ∈ ℚ ↦ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝑎 = ((1st ‘𝑥) / (2nd ‘𝑥)))))) | |
6 | fvex 6891 | . 2 ⊢ (2nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1st ‘𝑥) gcd (2nd ‘𝑥)) = 1 ∧ 𝐴 = ((1st ‘𝑥) / (2nd ‘𝑥))))) ∈ V | |
7 | 4, 5, 6 | fvmpt 6984 | 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 396 = wceq 1541 ∈ wcel 2106 × cxp 5667 ‘cfv 6532 ℩crio 7348 (class class class)co 7393 1st c1st 7955 2nd c2nd 7956 1c1 11093 / cdiv 11853 ℕcn 12194 ℤcz 12540 ℚcq 12914 gcd cgcd 16417 denomcdenom 16652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fv 6540 df-riota 7349 df-denom 16654 |
This theorem is referenced by: qnumdencl 16657 fden 16661 qnumdenbi 16662 |
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