qnumdencl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > qnumdencl		Structured version Visualization version GIF version

Theorem qnumdencl 16073

Description: Lemma for qnumcl 16074 and qdencl 16075. (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 15996	. . 3 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))
2		riotacl 7125	. . 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 7717	. . 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 16071	. . . . . . 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 2897	. . . . . 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 16072	. . . . . . 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 2897	. . . . . 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 250	. . . 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 493	. . 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	syl5bi 244	. 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 398 = wceq 1533 ∈ wcel 2110 ∃!wreu 3140 ⟨cop 4566 × cxp 5547 ‘cfv 6349 ℩crio 7107 (class class class)co 7150 1^st c1st 7681 2^nd c2nd 7682 1c1 10532 / cdiv 11291 ℕcn 11632 ℤcz 11975 ℚcq 12342 gcd cgcd 15837 numercnumer 16067 denomcdenom 16068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-q 12343 df-rp 12384 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-dvds 15602 df-gcd 15838 df-numer 16069 df-denom 16070

This theorem is referenced by: qnumcl 16074 qdencl 16075

W3C validator