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Theorem qnumdencl 12328

Description: Lemma for qnumcl 12329 and qdencl 12330. (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 12238	. . 3 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1^st ‘𝑎) gcd (2^nd ‘𝑎)) = 1 ∧ 𝐴 = ((1^st ‘𝑎) / (2^nd ‘𝑎))))
2		riotacl 5889	. . 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 14	. 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 6224	. . 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 12326	. . . . . . 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 2262	. . . . . 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 12327	. . . . . . 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 2262	. . . . . 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 473	. . . . 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 158	. . . 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 278	. . 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 152	. 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 13	1 ⊢ (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∃!wreu 2474 ⟨cop 3622 × cxp 4658 ‘cfv 5255 ℩crio 5873 (class class class)co 5919 1^st c1st 6193 2^nd c2nd 6194 1c1 7875 / cdiv 8693 ℕcn 8984 ℤcz 9320 ℚcq 9687 gcd cgcd 12082 numercnumer 12322 denomcdenom 12323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-sup 7045 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-fz 10078 df-fzo 10212 df-fl 10342 df-mod 10397 df-seqfrec 10522 df-exp 10613 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-dvds 11934 df-gcd 12083 df-numer 12324 df-denom 12325

This theorem is referenced by: qnumcl 12329 qdencl 12330

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