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Theorem qnumdencl 16710
Description: Lemma for qnumcl 16711 and qdencl 16712. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
qnumdencl (𝐴 ∈ β„š β†’ ((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•))

Proof of Theorem qnumdencl
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 qredeu 16628 . . 3 (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 7390 . . 3 (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„•))
31, 2syl 17 . 2 (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)))) ∈ (β„€ Γ— β„•))
4 elxp6 8025 . . 3 ((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„©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)))))⟩ ∧ ((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))))) ∈ β„€ ∧ (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))))) ∈ β„•)))
5 qnumval 16708 . . . . . . 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))))))
65eleq1d 2810 . . . . . 6 (𝐴 ∈ β„š β†’ ((numerβ€˜π΄) ∈ β„€ ↔ (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))))) ∈ β„€))
7 qdenval 16709 . . . . . . 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))))))
87eleq1d 2810 . . . . . 6 (𝐴 ∈ β„š β†’ ((denomβ€˜π΄) ∈ β„• ↔ (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))))) ∈ β„•))
96, 8anbi12d 630 . . . . 5 (𝐴 ∈ β„š β†’ (((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•) ↔ ((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))))) ∈ β„€ ∧ (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))))) ∈ β„•)))
109biimprd 247 . . . 4 (𝐴 ∈ β„š β†’ (((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))))) ∈ β„€ ∧ (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))))) ∈ β„•) β†’ ((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•)))
1110adantld 489 . . 3 (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)))))⟩ ∧ ((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))))) ∈ β„€ ∧ (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))))) ∈ β„•)) β†’ ((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•)))
124, 11biimtrid 241 . 2 (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)))) ∈ (β„€ Γ— β„•) β†’ ((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•)))
133, 12mpd 15 1 (𝐴 ∈ β„š β†’ ((numerβ€˜π΄) ∈ β„€ ∧ (denomβ€˜π΄) ∈ β„•))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒ!wreu 3362  βŸ¨cop 4630   Γ— cxp 5670  β€˜cfv 6543  β„©crio 7371  (class class class)co 7416  1st c1st 7989  2nd c2nd 7990  1c1 11139   / cdiv 11901  β„•cn 12242  β„€cz 12588  β„šcq 12962   gcd cgcd 16468  numercnumer 16704  denomcdenom 16705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-sup 9465  df-inf 9466  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-q 12963  df-rp 13007  df-fl 13789  df-mod 13867  df-seq 13999  df-exp 14059  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-dvds 16231  df-gcd 16469  df-numer 16706  df-denom 16707
This theorem is referenced by:  qnumcl  16711  qdencl  16712
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