qnumdencl - Metamath Proof Explorer

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Theorem qnumdencl 16836

Description: Lemma for qnumcl 16837 and qdencl 16838. (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 16754	. . 3 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))
2		riotacl 7391	. . 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 18	. 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 8024	. . 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 16834	. . . . . . 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 2847	. . . . . 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 16835	. . . . . . 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 2847	. . . . . 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 644	. . . . 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 251	. . . 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 496	. . 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 245	. 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 16	1 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃!wreu 3365 ⟨cop 4593 × cxp 5657 ‘cfv 6537 ℩crio 7373 (class class class)co 7417 1^st c1st 7988 2^nd c2nd 7989 1c1 11129 / cdiv 11899 ℕcn 12261 ℤcz 12619 ℚcq 13001 gcd cgcd 16590 numercnumer 16830 denomcdenom 16831

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-q 13002 df-rp 13047 df-fl 13857 df-mod 13935 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-dvds 16349 df-gcd 16591 df-numer 16832 df-denom 16833

This theorem is used by: qnumcl 16837 qdencl 16838

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