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Theorem qnumdenbi 16682
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 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢)) ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))

Proof of Theorem qnumdenbi
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 qredeu 16597 . . . . . . 7 (𝐴 ∈ β„š β†’ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))
2 riotacl 7385 . . . . . . 7 (βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))) β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) ∈ (β„€ Γ— β„•))
3 1st2nd2 8016 . . . . . . 7 ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) ∈ (β„€ Γ— β„•) β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
41, 2, 33syl 18 . . . . . 6 (𝐴 ∈ β„š β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
5 qnumval 16675 . . . . . . 7 (𝐴 ∈ β„š β†’ (numerβ€˜π΄) = (1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))))
6 qdenval 16676 . . . . . . 7 (𝐴 ∈ β„š β†’ (denomβ€˜π΄) = (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))))
75, 6opeq12d 4881 . . . . . 6 (𝐴 ∈ β„š β†’ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨(1st β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))), (2nd β€˜(β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))))⟩)
84, 7eqtr4d 2775 . . . . 5 (𝐴 ∈ β„š β†’ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩)
98eqeq1d 2734 . . . 4 (𝐴 ∈ β„š β†’ ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩ ↔ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩))
1093ad2ant1 1133 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩ ↔ ⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩))
11 fvex 6904 . . . 4 (numerβ€˜π΄) ∈ V
12 fvex 6904 . . . 4 (denomβ€˜π΄) ∈ V
1311, 12opth 5476 . . 3 (⟨(numerβ€˜π΄), (denomβ€˜π΄)⟩ = ⟨𝐡, 𝐢⟩ ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢))
1410, 13bitr2di 287 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢) ↔ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩))
15 opelxpi 5713 . . . 4 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
16153adant1 1130 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
1713ad2ant1 1133 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))))
18 fveq2 6891 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (1st β€˜π‘Ž) = (1st β€˜βŸ¨π΅, 𝐢⟩))
19 fveq2 6891 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (2nd β€˜π‘Ž) = (2nd β€˜βŸ¨π΅, 𝐢⟩))
2018, 19oveq12d 7429 . . . . . 6 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)))
2120eqeq1d 2734 . . . . 5 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ↔ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1))
2218, 19oveq12d 7429 . . . . . 6 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)) = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)))
2322eqeq2d 2743 . . . . 5 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)) ↔ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))))
2421, 23anbi12d 631 . . . 4 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ ((((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž))) ↔ (((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)))))
2524riota2 7393 . . 3 ((⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•) ∧ βˆƒ!π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩))
2616, 17, 25syl2anc 584 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ (β„©π‘Ž ∈ (β„€ Γ— β„•)(((1st β€˜π‘Ž) gcd (2nd β€˜π‘Ž)) = 1 ∧ 𝐴 = ((1st β€˜π‘Ž) / (2nd β€˜π‘Ž)))) = ⟨𝐡, 𝐢⟩))
27 op1stg 7989 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
28 op2ndg 7990 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
2927, 28oveq12d 7429 . . . . 5 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
30293adant1 1130 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
3130eqeq1d 2734 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ↔ (𝐡 gcd 𝐢) = 1))
32273adant1 1130 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
33283adant1 1130 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
3432, 33oveq12d 7429 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 / 𝐢))
3534eqeq2d 2743 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)) ↔ 𝐴 = (𝐡 / 𝐢)))
3631, 35anbi12d 631 . 2 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = 1 ∧ 𝐴 = ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩))) ↔ ((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢))))
3714, 26, 363bitr2rd 307 1 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (((𝐡 gcd 𝐢) = 1 ∧ 𝐴 = (𝐡 / 𝐢)) ↔ ((numerβ€˜π΄) = 𝐡 ∧ (denomβ€˜π΄) = 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒ!wreu 3374  βŸ¨cop 4634   Γ— cxp 5674  β€˜cfv 6543  β„©crio 7366  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  1c1 11113   / cdiv 11873  β„•cn 12214  β„€cz 12560  β„šcq 12934   gcd cgcd 16437  numercnumer 16671  denomcdenom 16672
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-q 12935  df-rp 12977  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-dvds 16200  df-gcd 16438  df-numer 16673  df-denom 16674
This theorem is referenced by:  qnumdencoprm  16683  qeqnumdivden  16684  divnumden  16686  numdensq  16692  numdenneg  32061  qqh0  33033  qqh1  33034  numdenexp  41310
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