Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  qnumdencl Structured version   Visualization version   GIF version

Theorem qnumdencl 16147
 Description: Lemma for qnumcl 16148 and qdencl 16149. (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 16067 . . 3 (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
2 riotacl 7131 . . 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 7733 . . 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 16145 . . . . . . 7 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
65eleq1d 2836 . . . . . 6 (𝐴 ∈ ℚ → ((numer‘𝐴) ∈ ℤ ↔ (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℤ))
7 qdenval 16146 . . . . . . 7 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
87eleq1d 2836 . . . . . 6 (𝐴 ∈ ℚ → ((denom‘𝐴) ∈ ℕ ↔ (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))) ∈ ℕ))
96, 8anbi12d 633 . . . . 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 251 . . . 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 494 . . 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 245 . 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 399   = wceq 1538   ∈ wcel 2111  ∃!wreu 3072  ⟨cop 4531   × cxp 5526  ‘cfv 6340  ℩crio 7113  (class class class)co 7156  1st c1st 7697  2nd c2nd 7698  1c1 10589   / cdiv 11348  ℕcn 11687  ℤcz 12033  ℚcq 12401   gcd cgcd 15906  numercnumer 16141  denomcdenom 16142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665  ax-pre-sup 10666 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-sup 8952  df-inf 8953  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-div 11349  df-nn 11688  df-2 11750  df-3 11751  df-n0 11948  df-z 12034  df-uz 12296  df-q 12402  df-rp 12444  df-fl 13224  df-mod 13300  df-seq 13432  df-exp 13493  df-cj 14519  df-re 14520  df-im 14521  df-sqrt 14655  df-abs 14656  df-dvds 15669  df-gcd 15907  df-numer 16143  df-denom 16144 This theorem is referenced by:  qnumcl  16148  qdencl  16149
 Copyright terms: Public domain W3C validator