Theorem qnumdenbi 16705

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 16618	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
2		riotacl 7334	. . . . . . 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 7974	. . . . . . 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 16698	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (numer‘𝐴) = (1st ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
6		qdenval 16699	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (denom‘𝐴) = (2nd ‘(℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))))
7	5, 6	opeq12d 4825	. . . . . 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 2775	. . . . 5 ⊢ (𝐴 ∈ ℚ → (℩𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)))) = ⟨(numer‘𝐴), (denom‘𝐴)⟩)
9	8	eqeq1d 2739	. . . 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 1134	. . 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 6847	. . . 4 ⊢ (numer‘𝐴) ∈ V
12		fvex 6847	. . . 4 ⊢ (denom‘𝐴) ∈ V
13	11, 12	opth 5424	. . 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 5661	. . . 4 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
16	15	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ⟨𝐵, 𝐶⟩ ∈ (ℤ × ℕ))
17	1	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ∃!𝑎 ∈ (ℤ × ℕ)(((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ∧ 𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎))))
18		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (1st ‘𝑎) = (1st ‘⟨𝐵, 𝐶⟩))
19		fveq2 6834	. . . . . . 7 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (2nd ‘𝑎) = (2nd ‘⟨𝐵, 𝐶⟩))
20	18, 19	oveq12d 7378	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) gcd (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)))
21	20	eqeq1d 2739	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (((1st ‘𝑎) gcd (2nd ‘𝑎)) = 1 ↔ ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1))
22	18, 19	oveq12d 7378	. . . . . 6 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → ((1st ‘𝑎) / (2nd ‘𝑎)) = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)))
23	22	eqeq2d 2748	. . . . 5 ⊢ (𝑎 = ⟨𝐵, 𝐶⟩ → (𝐴 = ((1st ‘𝑎) / (2nd ‘𝑎)) ↔ 𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩))))
24	21, 23	anbi12d 633	. . . 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 7342	. . 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 585	. 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 7947	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
28		op2ndg 7948	. . . . . 6 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
29	27, 28	oveq12d 7378	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
30	29	3adant1 1131	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 gcd 𝐶))
31	30	eqeq1d 2739	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (((1st ‘⟨𝐵, 𝐶⟩) gcd (2nd ‘⟨𝐵, 𝐶⟩)) = 1 ↔ (𝐵 gcd 𝐶) = 1))
32	27	3adant1 1131	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (1st ‘⟨𝐵, 𝐶⟩) = 𝐵)
33	28	3adant1 1131	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (2nd ‘⟨𝐵, 𝐶⟩) = 𝐶)
34	32, 33	oveq12d 7378	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) = (𝐵 / 𝐶))
35	34	eqeq2d 2748	. . 3 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℕ) → (𝐴 = ((1st ‘⟨𝐵, 𝐶⟩) / (2nd ‘⟨𝐵, 𝐶⟩)) ↔ 𝐴 = (𝐵 / 𝐶)))
36	31, 35	anbi12d 633	. 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 1087   = wceq 1542   ∈ wcel 2114  ∃!wreu 3341  ⟨cop 4574   × cxp 5622  ‘cfv 6492  ℩crio 7316  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934  1c1 11030   / cdiv 11798  ℕcn 12165  ℤcz 12515  ℚcq 12889   gcd cgcd 16454  numercnumer 16694  denomcdenom 16695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-q 12890  df-rp 12934  df-fl 13742  df-mod 13820  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-dvds 16213  df-gcd 16455  df-numer 16696  df-denom 16697

This theorem is referenced by:  qnumdencoprm  16706  qeqnumdivden  16707  divnumden  16709  numdensq  16715  numdenexp  16721  znumd  32901  zdend  32902  numdenneg  32903  qqh0  34144  qqh1  34145