Theorem qnumdenbi 15939

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 15857	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
2		riotacl 6950	. . . . . . 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 7539	. . . . . . 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 18	. . . . . 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 15932	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
6		qdenval 15933	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
7	5, 6	opeq12d 4682	. . . . . 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 2812	. . . . 5 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
9	8	eqeq1d 2775	. . . 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 1114	. . 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 6510	. . . 4 ⊢ (numer‘𝐴) ∈ V
12		fvex 6510	. . . 4 ⊢ (denom‘𝐴) ∈ V
13	11, 12	opth 5222	. . 3 ⊢ (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶))
14	10, 13	syl6rbb 280	. 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 5441	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16	15	3adant1 1111	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
17	1	3ad2ant1 1114	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
18		fveq2 6497	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (1st ‘𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19		fveq2 6497	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd ‘𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
20	18, 19	oveq12d 6993	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) gcd (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
21	20	eqeq1d 2775	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
22	18, 19	oveq12d 6993	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) / (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
23	22	eqeq2d 2783	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
24	21, 23	anbi12d 622	. . . 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 6958	. . 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 576	. 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 7512	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28		op2ndg 7513	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
29	27, 28	oveq12d 6993	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30	29	3adant1 1111	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
31	30	eqeq1d 2775	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32	27	3adant1 1111	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33	28	3adant1 1111	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
34	32, 33	oveq12d 6993	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
35	34	eqeq2d 2783	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
36	31, 35	anbi12d 622	. 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 300	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 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∃!wreu 3085  ⟨cop 4442   × cxp 5402  ‘cfv 6186  ℩crio 6935  (class class class)co 6975  1^st c1st 7498  2^nd c2nd 7499  1c1 10335   / cdiv 11097  ℕcn 11438  ℤcz 11792  ℚcq 12161   gcd cgcd 15702  numercnumer 15928  denomcdenom 15929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278  ax-cnex 10390  ax-resscn 10391  ax-1cn 10392  ax-icn 10393  ax-addcl 10394  ax-addrcl 10395  ax-mulcl 10396  ax-mulrcl 10397  ax-mulcom 10398  ax-addass 10399  ax-mulass 10400  ax-distr 10401  ax-i2m1 10402  ax-1ne0 10403  ax-1rid 10404  ax-rnegex 10405  ax-rrecex 10406  ax-cnre 10407  ax-pre-lttri 10408  ax-pre-lttrn 10409  ax-pre-ltadd 10410  ax-pre-mulgt0 10411  ax-pre-sup 10412

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-nel 3069  df-ral 3088  df-rex 3089  df-reu 3090  df-rmo 3091  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-pss 3840  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-tp 4441  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-tr 5028  df-id 5309  df-eprel 5314  df-po 5323  df-so 5324  df-fr 5363  df-we 5365  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-pred 5984  df-ord 6030  df-on 6031  df-lim 6032  df-suc 6033  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-f1 6191  df-fo 6192  df-f1o 6193  df-fv 6194  df-riota 6936  df-ov 6978  df-oprab 6979  df-mpo 6980  df-om 7396  df-1st 7500  df-2nd 7501  df-wrecs 7749  df-recs 7811  df-rdg 7849  df-er 8088  df-en 8306  df-dom 8307  df-sdom 8308  df-sup 8700  df-inf 8701  df-pnf 10475  df-mnf 10476  df-xr 10477  df-ltxr 10478  df-le 10479  df-sub 10671  df-neg 10672  df-div 11098  df-nn 11439  df-2 11502  df-3 11503  df-n0 11707  df-z 11793  df-uz 12058  df-q 12162  df-rp 12204  df-fl 12976  df-mod 13052  df-seq 13184  df-exp 13244  df-cj 14318  df-re 14319  df-im 14320  df-sqrt 14454  df-abs 14455  df-dvds 15467  df-gcd 15703  df-numer 15930  df-denom 15931

This theorem is referenced by:  qnumdencoprm  15940  qeqnumdivden  15941  divnumden  15943  numdensq  15949  numdenneg  30304  qqh0  30902  qqh1  30903  numdenexp  38652