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Theorem qnumval 15927

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

**Assertion**
Ref	Expression
qnumval	⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1^st ‘(℩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 qnumval Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2776	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 619	. . . 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 6933	. . 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 6497	. 2 ⊢ (𝑎 = 𝐴 → (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) = (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
5		df-numer 15925	. 2 ⊢ numer = (𝑎 ∈ ℚ ↦ (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))))
6		fvex 6506	. 2 ⊢ (1^st ‘(℩𝑥 ∈ (ℤ × ℕ)(((1^st ‘𝑥) gcd (2^nd ‘𝑥)) = 1 ∧ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))) ∈ V
7	4, 5, 6	fvmpt 6589	1 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1^st ‘(℩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 387 = wceq 1507 ∈ wcel 2050 × cxp 5399 ‘cfv 6182 ℩crio 6930 (class class class)co 6970 1^st c1st 7493 2^nd c2nd 7494 1c1 10330 / cdiv 11092 ℕcn 11433 ℤcz 11787 ℚcq 12156 gcd cgcd 15697 numercnumer 15923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-iota 6146 df-fun 6184 df-fv 6190 df-riota 6931 df-numer 15925

This theorem is referenced by: qnumdencl 15929 fnum 15932 qnumdenbi 15934

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