Theorem elq2 33403

Description: Elementhood in the rational numbers, providing the canonical representation. (Contributed by Thierry Arnoux, 9-Nov-2025.)

Assertion

Ref	Expression
elq2	⊢ (𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))

Distinct variable group:   𝑄,𝑝,𝑞

Proof of Theorem elq2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7427	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞))
2	1	eqeq2d 2772	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑄 = (𝑝 / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞)))
3		oveq1 7427	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞))
4	3	eqeq1d 2763	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → ((𝑝 gcd 𝑞) = 1 ↔ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1))
5	2, 4	anbi12d 644	. . 3 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → ((𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1) ↔ (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ∧ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1)))
6		oveq2 7428	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
7	6	eqeq2d 2772	. . . 4 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦)))))
8		oveq2 7428	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))))
9	8	eqeq1d 2763	. . . 4 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → (((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1 ↔ ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1))
10	7, 9	anbi12d 644	. . 3 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ∧ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1) ↔ (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) ∧ ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)))
11		simpllr 788	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℤ)
12		simplr 781	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℕ)
13	12	nnzd 12719	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
14	12	nnne0d 12388	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ≠ 0)
15		divgcdz 16683	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
16	11, 13, 14, 15	syl3anc 1398	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
17		divgcdnnr 16688	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
18	12, 11, 17	syl2anc 596	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
19		simpr 490	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = (𝑥 / 𝑦))
20	11	zcnd 12804	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
21	12	nncnd 12351	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℂ)
22	11, 13	gcdcld 16678	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℕ0)
23	22	nn0cnd 12669	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℂ)
24	14	neneqd 2961	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ 𝑦 = 0)
25	24	intnand 494	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
26		gcdeq0 16689	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
27	26	necon3abid 2992	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) ≠ 0 ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0)))
28	27	biimpar 483	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ¬ (𝑥 = 0 ∧ 𝑦 = 0)) → (𝑥 gcd 𝑦) ≠ 0)
29	11, 13, 25, 28	syl21anc 851	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ≠ 0)
30	20, 21, 23, 14, 29	divcan7d 12121	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) = (𝑥 / 𝑦))
31	19, 30	eqtr4d 2799	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
32		divgcdcoprm0 16840	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
33	11, 13, 14, 32	syl3anc 1398	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
34	31, 33	jca 521	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) ∧ ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1))
35	5, 10, 16, 18, 34	2rspcedvdw 3590	. 2 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))
36		elq 13077	. . 3 ⊢ (𝑄 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
37	36	biimpi 219	. 2 ⊢ (𝑄 ∈ ℚ → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
38	35, 37	r19.29vva 3223	1 ⊢ (𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  (class class class)co 7420  0cc0 11200  1c1 11201   / cdiv 11973  ℕcn 12335  ℤcz 12693  ℚcq 13075   gcd cgcd 16664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-z 12694  df-uz 12966  df-q 13076  df-rp 13121  df-fl 13932  df-mod 14010  df-seq 14145  df-exp 14205  df-cj 15266  df-re 15267  df-im 15268  df-sqrt 15402  df-abs 15403  df-dvds 16423  df-gcd 16665

This theorem is used by:  cos9thpiminplylem2  34415