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Theorem qdenval 12716

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 2236	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 464	. . . 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 5962	. . 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 5633	. 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 12714	. 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		zex 9463	. . . 4 ⊢ ℤ ∈ V
7		nnex 9124	. . . 4 ⊢ ℕ ∈ V
8	6, 7	xpex 4834	. . 3 ⊢ (ℤ × ℕ) ∈ V
9		riotaexg 5964	. . 3 ⊢ ((ℤ × ℕ) ∈ V → (℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))) ∈ V)
10		2ndexg 6320	. . 3 ⊢ ((℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥)))) ∈ V → (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) ∈ V)
11	8, 9, 10	mp2b 8	. 2 ⊢ (2^nd ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) ∈ V
12	4, 5, 11	fvmpt 5713	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 set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 Vcvv 2799 × cxp 4717 ‘cfv 5318 ℩crio 5959 (class class class)co 6007 1^st c1st 6290 2^nd c2nd 6291 1c1 8008 / cdiv 8827 ℕcn 9118 ℤcz 9454 ℚcq 9822 gcd cgcd 12482 denomcdenom 12712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-cnex 8098 ax-resscn 8099 ax-1re 8101 ax-addrcl 8104

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-fo 5324 df-fv 5326 df-riota 5960 df-ov 6010 df-2nd 6293 df-neg 8328 df-inn 9119 df-z 9455 df-denom 12714

This theorem is referenced by: qnumdencl 12717 fden 12721 qnumdenbi 12722

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