qdenval - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > qdenval		GIF version

Theorem qdenval 11046

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 2091	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 = ((1^st ‘𝑥) / (2^nd ‘𝑥)) ↔ 𝐴 = ((1^st ‘𝑥) / (2^nd ‘𝑥))))
2	1	anbi2d 452	. . . 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 5571	. . 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 5272	. 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 11044	. 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 8692	. . . 4 ⊢ ℤ ∈ V
7		nnex 8363	. . . 4 ⊢ ℕ ∈ V
8	6, 7	xpex 4521	. . 3 ⊢ (ℤ × ℕ) ∈ V
9		riotaexg 5573	. . 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 5896	. . 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 5344	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 102 = wceq 1287 ∈ wcel 1436 Vcvv 2615 × cxp 4409 ‘cfv 4981 ℩crio 5568 (class class class)co 5613 1^st c1st 5866 2^nd c2nd 5867 1c1 7295 / cdiv 8078 ℕcn 8357 ℤcz 8683 ℚcq 9036 gcd cgcd 10820 denomcdenom 11042

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3932 ax-pow 3984 ax-pr 4010 ax-un 4234 ax-cnex 7380 ax-resscn 7381 ax-1re 7383 ax-addrcl 7386

This theorem depends on definitions: df-bi 115 df-3or 923 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-rab 2364 df-v 2617 df-sbc 2830 df-un 2992 df-in 2994 df-ss 3001 df-pw 3417 df-sn 3437 df-pr 3438 df-op 3440 df-uni 3637 df-int 3672 df-br 3821 df-opab 3875 df-mpt 3876 df-id 4094 df-xp 4417 df-rel 4418 df-cnv 4419 df-co 4420 df-dm 4421 df-rn 4422 df-iota 4946 df-fun 4983 df-fn 4984 df-f 4985 df-fo 4987 df-fv 4989 df-riota 5569 df-ov 5616 df-2nd 5869 df-neg 7600 df-inn 8358 df-z 8684 df-denom 11044

This theorem is referenced by: qnumdencl 11047 fden 11051 qnumdenbi 11052

W3C validator