Theorem qnumdenbi 16768

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 16682	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
2		riotacl 7384	. . . . . . 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 8032	. . . . . . 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 16761	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
6		qdenval 16762	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
7	5, 6	opeq12d 4862	. . . . . 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 2774	. . . . 5 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
9	8	eqeq1d 2738	. . . 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 1133	. . 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 6894	. . . 4 ⊢ (numer‘𝐴) ∈ V
12		fvex 6894	. . . 4 ⊢ (denom‘𝐴) ∈ V
13	11, 12	opth 5456	. . 3 ⊢ (⟨(numer‘𝐴), (denom‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((numer‘𝐴) = 𝐵 ∧ (denom‘𝐴) = 𝐶))
14	10, 13	bitr2di 288	. 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 5696	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16	15	3adant1 1130	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
17	1	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
18		fveq2 6881	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (1st ‘𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19		fveq2 6881	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd ‘𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
20	18, 19	oveq12d 7428	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) gcd (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
21	20	eqeq1d 2738	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
22	18, 19	oveq12d 7428	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) / (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
23	22	eqeq2d 2747	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
24	21, 23	anbi12d 632	. . . 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 7392	. . 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 584	. 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 8005	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28		op2ndg 8006	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
29	27, 28	oveq12d 7428	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30	29	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
31	30	eqeq1d 2738	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32	27	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33	28	3adant1 1130	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
34	32, 33	oveq12d 7428	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
35	34	eqeq2d 2747	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
36	31, 35	anbi12d 632	. 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 308	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃!wreu 3362  ⟨cop 4612   × cxp 5657  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  1c1 11135   / cdiv 11899  ℕcn 12245  ℤcz 12593  ℚcq 12969   gcd cgcd 16518  numercnumer 16757  denomcdenom 16758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-q 12970  df-rp 13014  df-fl 13814  df-mod 13892  df-seq 14025  df-exp 14085  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-dvds 16278  df-gcd 16519  df-numer 16759  df-denom 16760

This theorem is referenced by:  qnumdencoprm  16769  qeqnumdivden  16770  divnumden  16772  numdensq  16778  numdenexp  16784  znumd  32796  zdend  32797  numdenneg  32798  qqh0  34020  qqh1  34021