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Theorem qnumdenbi 16685
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 16600 . . . . . . 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 7386 . . . . . . 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 8017 . . . . . . 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 16678 . . . . . . 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 16679 . . . . . . 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 4882 . . . . . 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 2774 . . . . 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 2733 . . . 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 1132 . . 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 6905 . . . 4 (numerβ€˜π΄) ∈ V
12 fvex 6905 . . . 4 (denomβ€˜π΄) ∈ V
1311, 12opth 5477 . . 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 5714 . . . 4 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
16153adant1 1129 . . 3 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ⟨𝐡, 𝐢⟩ ∈ (β„€ Γ— β„•))
1713ad2ant1 1132 . . 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 6892 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (1st β€˜π‘Ž) = (1st β€˜βŸ¨π΅, 𝐢⟩))
19 fveq2 6892 . . . . . . 7 (π‘Ž = ⟨𝐡, 𝐢⟩ β†’ (2nd β€˜π‘Ž) = (2nd β€˜βŸ¨π΅, 𝐢⟩))
2018, 19oveq12d 7430 . . . . . 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 2733 . . . . 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 7430 . . . . . 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 2742 . . . . 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 630 . . . 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 7394 . . 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 583 . 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 7990 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
28 op2ndg 7991 . . . . . 6 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
2927, 28oveq12d 7430 . . . . 5 ((𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
30293adant1 1129 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) gcd (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 gcd 𝐢))
3130eqeq1d 2733 . . 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 1129 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (1st β€˜βŸ¨π΅, 𝐢⟩) = 𝐡)
33283adant1 1129 . . . . 5 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ (2nd β€˜βŸ¨π΅, 𝐢⟩) = 𝐢)
3432, 33oveq12d 7430 . . . 4 ((𝐴 ∈ β„š ∧ 𝐡 ∈ β„€ ∧ 𝐢 ∈ β„•) β†’ ((1st β€˜βŸ¨π΅, 𝐢⟩) / (2nd β€˜βŸ¨π΅, 𝐢⟩)) = (𝐡 / 𝐢))
3534eqeq2d 2742 . . 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 630 . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆƒ!wreu 3373  βŸ¨cop 4635   Γ— cxp 5675  β€˜cfv 6544  β„©crio 7367  (class class class)co 7412  1st c1st 7976  2nd c2nd 7977  1c1 11114   / cdiv 11876  β„•cn 12217  β„€cz 12563  β„šcq 12937   gcd cgcd 16440  numercnumer 16674  denomcdenom 16675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9440  df-inf 9441  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-n0 12478  df-z 12564  df-uz 12828  df-q 12938  df-rp 12980  df-fl 13762  df-mod 13840  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-dvds 16203  df-gcd 16441  df-numer 16676  df-denom 16677
This theorem is referenced by:  qnumdencoprm  16686  qeqnumdivden  16687  divnumden  16689  numdensq  16695  numdenneg  32287  qqh0  33259  qqh1  33260  numdenexp  41531
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