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Theorem qdenval 16442

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 2742	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 629	. . . 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 7234	. . 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 6778	. 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 16440	. 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 6787	. 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 6875	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 396 = wceq 1539 ∈ wcel 2106 × cxp 5587 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 1c1 10872 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℚcq 12688 gcd cgcd 16201 denomcdenom 16438

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-riota 7232 df-denom 16440

This theorem is referenced by: qnumdencl 16443 fden 16447 qnumdenbi 16448

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