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Theorem qnumdenbi 16676
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 16591 . . . . . . 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 7379 . . . . . . 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 8010 . . . . . . 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 16669 . . . . . . 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 16670 . . . . . . 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 4880 . . . . . 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 6901 . . . 4 (numerβ€˜π΄) ∈ V
12 fvex 6901 . . . 4 (denomβ€˜π΄) ∈ V
1311, 12opth 5475 . . 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 5712 . . . 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 6888 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (1st β€˜π‘Ž) = (1st β€˜βŸ¨π΅, 𝐢⟩))
19 fveq2 6888 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (2nd β€˜π‘Ž) = (2nd β€˜βŸ¨π΅, 𝐢⟩))
2018, 19oveq12d 7423 . . . . . 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 7423 . . . . . 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 7387 . . 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 7983 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
28 op2ndg 7984 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
2927, 28oveq12d 7423 . . . . 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 7423 . . . 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 4633   Γ— cxp 5673  β€˜cfv 6540  β„©crio 7360  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  1c1 11107   / cdiv 11867  β„•cn 12208  β„€cz 12554  β„šcq 12928   gcd cgcd 16431  numercnumer 16665  denomcdenom 16666
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 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-fl 13753  df-mod 13831  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-dvds 16194  df-gcd 16432  df-numer 16667  df-denom 16668
This theorem is referenced by:  qnumdencoprm  16677  qeqnumdivden  16678  divnumden  16680  numdensq  16686  numdenneg  32010  qqh0  32952  qqh1  32953  numdenexp  41223
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