Theorem elq2 32841

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 7363	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞))
2	1	eqeq2d 2745	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑄 = (𝑝 / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞)))
3		oveq1 7363	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞))
4	3	eqeq1d 2736	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → ((𝑝 gcd 𝑞) = 1 ↔ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1))
5	2, 4	anbi12d 632	. . 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 7364	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
7	6	eqeq2d 2745	. . . 4 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦)))))
8		oveq2 7364	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))))
9	8	eqeq1d 2736	. . . 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 632	. . 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 775	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℤ)
12		simplr 768	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℕ)
13	12	nnzd 12512	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
14	12	nnne0d 12193	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ≠ 0)
15		divgcdz 16436	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
16	11, 13, 14, 15	syl3anc 1373	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
17		divgcdnnr 16441	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
18	12, 11, 17	syl2anc 584	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
19		simpr 484	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = (𝑥 / 𝑦))
20	11	zcnd 12595	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
21	12	nncnd 12159	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℂ)
22	11, 13	gcdcld 16433	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℕ0)
23	22	nn0cnd 12462	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℂ)
24	14	neneqd 2935	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ 𝑦 = 0)
25	24	intnand 488	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
26		gcdeq0 16442	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
27	26	necon3abid 2966	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) ≠ 0 ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0)))
28	27	biimpar 477	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ¬ (𝑥 = 0 ∧ 𝑦 = 0)) → (𝑥 gcd 𝑦) ≠ 0)
29	11, 13, 25, 28	syl21anc 837	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ≠ 0)
30	20, 21, 23, 14, 29	divcan7d 11943	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) = (𝑥 / 𝑦))
31	19, 30	eqtr4d 2772	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
32		divgcdcoprm0 16590	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
33	11, 13, 14, 32	syl3anc 1373	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
34	31, 33	jca 511	. . 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 3588	. 2 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))
36		elq 12861	. . 3 ⊢ (𝑄 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
37	36	biimpi 216	. 2 ⊢ (𝑄 ∈ ℚ → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
38	35, 37	r19.29vva 3194	1 ⊢ (𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∃wrex 3058  (class class class)co 7356  0cc0 11024  1c1 11025   / cdiv 11792  ℕcn 12143  ℤcz 12486  ℚcq 12859   gcd cgcd 16419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-er 8633  df-en 8882  df-dom 8883  df-sdom 8884  df-sup 9343  df-inf 9344  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-n0 12400  df-z 12487  df-uz 12750  df-q 12860  df-rp 12904  df-fl 13710  df-mod 13788  df-seq 13923  df-exp 13983  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-dvds 16178  df-gcd 16420

This theorem is referenced by:  cos9thpiminplylem2  33889