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Theorem qnumdencl 16763

Description: Lemma for qnumcl 16764 and qdencl 16765. (Contributed by Stefan O'Rear, 13-Sep-2014.)

**Assertion**
Ref	Expression
qnumdencl	⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Proof of Theorem qnumdencl Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qredeu 16682	. . 3 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))
2		riotacl 7384	. . 3 ⊢ (∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))) → (℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) ∈ (ℤ × ℕ))
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) ∈ (ℤ × ℕ))
4		elxp6 8027	. . 3 ⊢ ((℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) ∈ (ℤ × ℕ) ↔ ((℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) = ⟨(1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))), (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))))⟩ ∧ ((1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℤ ∧ (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℕ)))
5		qnumval 16761	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))))
6	5	eleq1d 2820	. . . . . 6 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ↔ (1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℤ))
7		qdenval 16762	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))))
8	7	eleq1d 2820	. . . . . 6 ⊢ (𝐴 ∈ ℚ → ((denom‘𝐴) ∈ ℕ ↔ (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℕ))
9	6, 8	anbi12d 632	. . . . 5 ⊢ (𝐴 ∈ ℚ → (((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ) ↔ ((1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℤ ∧ (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℕ)))
10	9	biimprd 248	. . . 4 ⊢ (𝐴 ∈ ℚ → (((1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℤ ∧ (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℕ) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
11	10	adantld 490	. . 3 ⊢ (𝐴 ∈ ℚ → (((℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) = ⟨(1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))), (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))))⟩ ∧ ((1^st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℤ ∧ (2^nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))) ∈ ℕ)) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
12	4, 11	biimtrid 242	. 2 ⊢ (𝐴 ∈ ℚ → ((℩𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎)))) ∈ (ℤ × ℕ) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
13	3, 12	mpd 15	1 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃!wreu 3362 ⟨cop 4612 × cxp 5657 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 1c1 11135 / cdiv 11899 ℕcn 12245 ℤcz 12593 ℚcq 12969 gcd cgcd 16518 numercnumer 16757 denomcdenom 16758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-q 12970 df-rp 13014 df-fl 13814 df-mod 13892 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-dvds 16278 df-gcd 16519 df-numer 16759 df-denom 16760

This theorem is referenced by: qnumcl 16764 qdencl 16765

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