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Theorem qnumdencl 16668

Description: Lemma for qnumcl 16669 and qdencl 16670. (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 16587	. . 3 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))
2		riotacl 7327	. . 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 7965	. . 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 16666	. . . . . . 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 2813	. . . . . 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 16667	. . . . . . 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 2813	. . . . . 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 3343 ⟨cop 4585 × cxp 5621 ‘cfv 6486 ℩crio 7309 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 1c1 11029 / cdiv 11795 ℕcn 12146 ℤcz 12489 ℚcq 12867 gcd cgcd 16423 numercnumer 16662 denomcdenom 16663

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 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-q 12868 df-rp 12912 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424 df-numer 16664 df-denom 16665

This theorem is referenced by: qnumcl 16669 qdencl 16670

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