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Theorem qnumdencl 16621
Description: Lemma for qnumcl 16622 and qdencl 16623. (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 16541 . . 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 7336 . . 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 7960 . . 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 16619 . . . . . . 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 2823 . . . . . 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 16620 . . . . . . 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 2823 . . . . . 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 632 . . . . 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 248 . . . 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 492 . . 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 397   = wceq 1542   ∈ wcel 2107  βˆƒ!wreu 3354  βŸ¨cop 4597   Γ— cxp 5636  β€˜cfv 6501  β„©crio 7317  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  1c1 11059   / cdiv 11819  β„•cn 12160  β„€cz 12506  β„šcq 12880   gcd cgcd 16381  numercnumer 16615  denomcdenom 16616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-3 12224  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-fl 13704  df-mod 13782  df-seq 13914  df-exp 13975  df-cj 14991  df-re 14992  df-im 14993  df-sqrt 15127  df-abs 15128  df-dvds 16144  df-gcd 16382  df-numer 16617  df-denom 16618
This theorem is referenced by:  qnumcl  16622  qdencl  16623
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