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Theorem qnumdencl 15285
 Description: Lemma for qnumcl 15286 and qdencl 15287. (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 15210 . . 3 (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
2 riotacl 6525 . . 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 17 . 2 (𝐴 ∈ ℚ → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ))
4 elxp6 7091 . . 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 15283 . . . . . . 7 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
65eleq1d 2672 . . . . . 6 (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ↔ (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ))
7 qdenval 15284 . . . . . . 7 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
87eleq1d 2672 . . . . . 6 (𝐴 ∈ ℚ → ((denom‘𝐴) ∈ ℕ ↔ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ))
96, 8anbi12d 743 . . . . 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 237 . . . 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 482 . . 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 231 . 2 (𝐴 ∈ ℚ → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ) → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ)))
133, 12mpd 15 1 (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ∧ (denom‘𝐴) ∈ ℕ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  ⟨cop 4131   × cxp 5036  ‘cfv 5804  ℩crio 6510  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  1c1 9816   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℚcq 11664   gcd cgcd 15054  numercnumer 15279  denomcdenom 15280 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055  df-numer 15281  df-denom 15282 This theorem is referenced by:  qnumcl  15286  qdencl  15287
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