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Theorem qdenval 16680

Description: Value of the canonical denominator function. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
qdenval	⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Distinct variable group: 𝑥,𝐴

Proof of Theorem qdenval Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2730	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 628	. . . 4 ⊢ (𝑎 = 𝐴 → ((((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥))) ↔ (((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))))
3	2	riotabidv 7362	. . 3 ⊢ (𝑎 = 𝐴 → (℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))) = (℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))))
4	3	fveq2d 6888	. 2 ⊢ (𝑎 = 𝐴 → (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) = (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
5		df-denom 16678	. 2 ⊢ denom = (𝑎 ∈ ℚ ↦ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
6		fvex 6897	. 2 ⊢ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) ∈ V
7	4, 5, 6	fvmpt 6991	1 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 × cxp 5667 ‘cfv 6536 ℩crio 7359 (class class class)co 7404 1^st c1st 7969 2^nd c2nd 7970 1c1 11110 / cdiv 11872 ℕcn 12213 ℤcz 12559 ℚcq 12933 gcd cgcd 16439 denomcdenom 16676

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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420

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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6488 df-fun 6538 df-fv 6544 df-riota 7360 df-denom 16678

This theorem is referenced by: qnumdencl 16681 fden 16685 qnumdenbi 16686

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