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Theorem qnumdencl 12078
Description: Lemma for qnumcl 12079 and qdencl 12080. (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.
StepHypRef Expression
1 qredeu 11990 . . 3 (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
2 riotacl 5797 . . 3 (∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))) → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ))
31, 2syl 14 . 2 (𝐴 ∈ ℚ → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ))
4 elxp6 6120 . . 3 ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ) ↔ ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩ ∧ ((1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ ∧ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ)))
5 qnumval 12076 . . . . . . 7 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
65eleq1d 2226 . . . . . 6 (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ↔ (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ))
7 qdenval 12077 . . . . . . 7 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
87eleq1d 2226 . . . . . 6 (𝐴 ∈ ℚ → ((denom‘𝐴) ∈ ℕ ↔ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ))
96, 8anbi12d 465 . . . . 5 (𝐴 ∈ ℚ → (((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ) ↔ ((1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ ∧ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ)))
109biimprd 157 . . . 4 (𝐴 ∈ ℚ → (((1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ ∧ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
1110adantld 276 . . 3 (𝐴 ∈ ℚ → (((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩ ∧ ((1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ ∧ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ)) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
124, 11syl5bi 151 . 2 (𝐴 ∈ ℚ → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
133, 12mpd 13 1 (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  ∃!wreu 2437  cop 3564   × cxp 4587  cfv 5173  crio 5782  (class class class)co 5827  1st c1st 6089  2nd c2nd 6090  1c1 7736   / cdiv 8550  cn 8839  cz 9173  cq 9535   gcd cgcd 11842  numercnumer 12072  denomcdenom 12073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550  ax-cnex 7826  ax-resscn 7827  ax-1cn 7828  ax-1re 7829  ax-icn 7830  ax-addcl 7831  ax-addrcl 7832  ax-mulcl 7833  ax-mulrcl 7834  ax-addcom 7835  ax-mulcom 7836  ax-addass 7837  ax-mulass 7838  ax-distr 7839  ax-i2m1 7840  ax-0lt1 7841  ax-1rid 7842  ax-0id 7843  ax-rnegex 7844  ax-precex 7845  ax-cnre 7846  ax-pre-ltirr 7847  ax-pre-ltwlin 7848  ax-pre-lttrn 7849  ax-pre-apti 7850  ax-pre-ltadd 7851  ax-pre-mulgt0 7852  ax-pre-mulext 7853  ax-arch 7854  ax-caucvg 7855
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-if 3507  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-ilim 4332  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-riota 5783  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-frec 6341  df-sup 6931  df-pnf 7917  df-mnf 7918  df-xr 7919  df-ltxr 7920  df-le 7921  df-sub 8053  df-neg 8054  df-reap 8455  df-ap 8462  df-div 8551  df-inn 8840  df-2 8898  df-3 8899  df-4 8900  df-n0 9097  df-z 9174  df-uz 9446  df-q 9536  df-rp 9568  df-fz 9920  df-fzo 10052  df-fl 10179  df-mod 10232  df-seqfrec 10355  df-exp 10429  df-cj 10754  df-re 10755  df-im 10756  df-rsqrt 10910  df-abs 10911  df-dvds 11696  df-gcd 11843  df-numer 12074  df-denom 12075
This theorem is referenced by:  qnumcl  12079  qdencl  12080
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