Theorem qnumdenbi 16925

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 16838	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
2		riotacl 7394	. . . . . . 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 8040	. . . . . . 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 16918	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
6		qdenval 16919	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
7	5, 6	opeq12d 4847	. . . . . 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 2798	. . . . 5 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
9	8	eqeq1d 2762	. . . 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 1151	. . 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 6898	. . . 4 ⊢ (numer‘𝐴) ∈ V
12		fvex 6898	. . . 4 ⊢ (denom‘𝐴) ∈ V
13	11, 12	opth 5444	. . 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 5687	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16	15	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
17	1	3ad2ant1 1151	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
18		fveq2 6885	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (1st ‘𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19		fveq2 6885	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd ‘𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
20	18, 19	oveq12d 7438	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) gcd (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
21	20	eqeq1d 2762	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
22	18, 19	oveq12d 7438	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) / (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
23	22	eqeq2d 2771	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
24	21, 23	anbi12d 644	. . . 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 7402	. . 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 596	. 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 8013	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28		op2ndg 8014	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
29	27, 28	oveq12d 7438	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30	29	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
31	30	eqeq1d 2762	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32	27	3adant1 1148	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33	28	3adant1 1148	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
34	32, 33	oveq12d 7438	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
35	34	eqeq2d 2771	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
36	31, 35	anbi12d 644	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∃!wreu 3363  ⟨cop 4834   × cxp 5648  ‘cfv 6538  ℩crio 7376  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  1c1 11204   / cdiv 11976  ℕcn 12338  ℤcz 12696  ℚcq 13078   gcd cgcd 16669  numercnumer 16914  denomcdenom 16915

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280  ax-pre-sup 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-sup 9437  df-inf 9438  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-div 11977  df-nn 12339  df-2 12408  df-3 12409  df-n0 12610  df-z 12697  df-uz 12969  df-q 13079  df-rp 13124  df-fl 13935  df-mod 14013  df-seq 14148  df-exp 14208  df-cj 15271  df-re 15272  df-im 15273  df-sqrt 15407  df-abs 15408  df-dvds 16428  df-gcd 16670  df-numer 16916  df-denom 16917

This theorem is used by:  qnumdencoprm  16926  qeqnumdivden  16927  divnumden  16929  numdensq  16935  numdenexp  16942  znumd  33450  zdend  33451  numdenneg  33452  qqh0  34662  qqh1  34663