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Theorem qnumdenbi 16300
Description: Two numbers are the canonical representation of a rational iff they are coprime and have the right quotient. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdenbi ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))

Proof of Theorem qnumdenbi
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 qredeu 16215 . . . . . . 7 (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
2 riotacl 7188 . . . . . . 7 (∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))) → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) ∈ (ℤ × ℕ))
3 1st2nd2 7800 . . . . . . 7 ((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)))))⟩)
41, 2, 33syl 18 . . . . . 6 (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)))))⟩)
5 qnumval 16293 . . . . . . 7 (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
6 qdenval 16294 . . . . . . 7 (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))))
75, 6opeq12d 4792 . . . . . 6 (𝐴 ∈ ℚ → ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨(1st ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))), (2nd ‘(𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))))⟩)
84, 7eqtr4d 2780 . . . . 5 (𝐴 ∈ ℚ → (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
98eqeq1d 2739 . . . 4 (𝐴 ∈ ℚ → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
1093ad2ant1 1135 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
11 fvex 6730 . . . 4 (numer‘𝐴) ∈ V
12 fvex 6730 . . . 4 (denom‘𝐴) ∈ V
1311, 12opth 5360 . . 3 (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶))
1410, 13bitr2di 291 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶) ↔ (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩))
15 opelxpi 5588 . . . 4 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16153adant1 1132 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
1713ad2ant1 1135 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))))
18 fveq2 6717 . . . . . . 7 (𝑎 = ⟨𝐵, 𝐶⟩ → (1st𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19 fveq2 6717 . . . . . . 7 (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
2018, 19oveq12d 7231 . . . . . 6 (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st𝑎) gcd (2nd𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
2120eqeq1d 2739 . . . . 5 (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st𝑎) gcd (2nd𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
2218, 19oveq12d 7231 . . . . . 6 (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st𝑎) / (2nd𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
2322eqeq2d 2748 . . . . 5 (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st𝑎) / (2nd𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
2421, 23anbi12d 634 . . . 4 (𝑎 = ⟨𝐵, 𝐶⟩ → ((((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎))) ↔ (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))))
2524riota2 7196 . . 3 ((⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ) ∧ ∃!𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩))
2616, 17, 25syl2anc 587 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (𝑎 ∈ (ℤ × ℕ)(((1st𝑎) gcd (2nd𝑎)) = 1 ∧ 𝐴 = ((1st𝑎) / (2nd𝑎)))) = ⟨𝐵, 𝐶⟩))
27 op1stg 7773 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28 op2ndg 7774 . . . . . 6 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
2927, 28oveq12d 7231 . . . . 5 ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30293adant1 1132 . . . 4 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
3130eqeq1d 2739 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32273adant1 1132 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33283adant1 1132 . . . . 5 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
3432, 33oveq12d 7231 . . . 4 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
3534eqeq2d 2748 . . 3 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
3631, 35anbi12d 634 . 2 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ ((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶))))
3714, 26, 363bitr2rd 311 1 ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  ∃!wreu 3063  cop 4547   × cxp 5549  cfv 6380  crio 7169  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  1c1 10730   / cdiv 11489  cn 11830  cz 12176  cq 12544   gcd cgcd 16053  numercnumer 16289  denomcdenom 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-fl 13367  df-mod 13443  df-seq 13575  df-exp 13636  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-dvds 15816  df-gcd 16054  df-numer 16291  df-denom 16292
This theorem is referenced by:  qnumdencoprm  16301  qeqnumdivden  16302  divnumden  16304  numdensq  16310  numdenneg  30851  qqh0  31646  qqh1  31647  numdenexp  40045
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