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Theorem qdenval 16651

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 2737	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 630	. . . 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 7311	. . 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 6832	. 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 16649	. 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 6841	. 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 6935	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 1541 ∈ wcel 2113 × cxp 5617 ‘cfv 6486 ℩crio 7308 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 1c1 11014 / cdiv 11781 ℕcn 12132 ℤcz 12475 ℚcq 12848 gcd cgcd 16407 denomcdenom 16647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6442 df-fun 6488 df-fv 6494 df-riota 7309 df-denom 16649

This theorem is referenced by: qnumdencl 16652 fden 16656 qnumdenbi 16657

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