Theorem qnumdenbi 16793

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 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))

Proof of Theorem qnumdenbi

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qredeu 16706	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
2		riotacl 7374	. . . . . . 7 ⊢ (∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))) → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) ∈ (ℤ × ℕ))
3		1st2nd2 8013	. . . . . . 7 ⊢ ((℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) ∈ (ℤ × ℕ) → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))), (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))))⟩)
4	1, 2, 3	3syl 19	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))), (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))))⟩)
5		qnumval 16786	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
6		qdenval 16787	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
7	5, 6	opeq12d 4842	. . . . . 6 ⊢ (𝐴 ∈ ℚ → ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨(1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))), (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))))⟩)
8	4, 7	eqtr4d 2803	. . . . 5 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
9	8	eqeq1d 2767	. . . 4 ⊢ (𝐴 ∈ ℚ → ((℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
10	9	3ad2ant1 1149	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨𝐵, 𝐶⟩ ↔ ⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
11		fvex 6884	. . . 4 ⊢ (numer‘𝐴) ∈ V
12		fvex 6884	. . . 4 ⊢ (denom‘𝐴) ∈ V
13	11, 12	opth 5449	. . 3 ⊢ (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶))
14	10, 13	bitr2di 291	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶) ↔ (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨𝐵, 𝐶⟩))
15		opelxpi 5689	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16	15	3adant1 1146	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
17	1	3ad2ant1 1149	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
18		fveq2 6871	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (1st ‘𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19		fveq2 6871	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd ‘𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
20	18, 19	oveq12d 7418	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) gcd (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
21	20	eqeq1d 2767	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
22	18, 19	oveq12d 7418	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) / (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
23	22	eqeq2d 2776	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
24	21, 23	anbi12d 643	. . . 4 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))) ↔ (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))))
25	24	riota2 7382	. . 3 ⊢ ((⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ) ∧ ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨𝐵, 𝐶⟩))
26	16, 17, 25	syl2anc 595	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨𝐵, 𝐶⟩))
27		op1stg 7986	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28		op2ndg 7987	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
29	27, 28	oveq12d 7418	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30	29	3adant1 1146	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
31	30	eqeq1d 2767	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32	27	3adant1 1146	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33	28	3adant1 1146	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
34	32, 33	oveq12d 7418	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
35	34	eqeq2d 2776	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
36	31, 35	anbi12d 643	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ∧ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))) ↔ ((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶))))
37	14, 26, 36	3bitr2rd 311	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((𝐵 gcd 𝐶) = 1 ∧ 𝐴 = (𝐵 / 𝐶)) ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃!wreu 3368  ⟨cop 4591   × cxp 5650  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  1c1 11089   / cdiv 11859  ℕcn 12224  ℤcz 12582  ℚcq 12963   gcd cgcd 16542  numercnumer 16782  denomcdenom 16783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-rp 13008  df-fl 13816  df-mod 13894  df-seq 14029  df-exp 14089  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-dvds 16301  df-gcd 16543  df-numer 16784  df-denom 16785

This theorem is referenced by:  qnumdencoprm  16794  qeqnumdivden  16795  divnumden  16797  numdensq  16803  numdenexp  16809  znumd  33070  zdend  33071  numdenneg  33072  qqh0  34291  qqh1  34292