Theorem elq2 33014

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 7403	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞))
2	1	eqeq2d 2773	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑄 = (𝑝 / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞)))
3		oveq1 7403	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞))
4	3	eqeq1d 2764	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → ((𝑝 gcd 𝑞) = 1 ↔ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1))
5	2, 4	anbi12d 641	. . 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 7404	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
7	6	eqeq2d 2773	. . . 4 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦)))))
8		oveq2 7404	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))))
9	8	eqeq1d 2764	. . . 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 641	. . 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 785	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℤ)
12		simplr 778	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℕ)
13	12	nnzd 12594	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
14	12	nnne0d 12263	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ≠ 0)
15		divgcdz 16545	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
16	11, 13, 14, 15	syl3anc 1390	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
17		divgcdnnr 16550	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
18	12, 11, 17	syl2anc 593	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
19		simpr 488	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = (𝑥 / 𝑦))
20	11	zcnd 12678	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
21	12	nncnd 12226	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℂ)
22	11, 13	gcdcld 16542	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℕ0)
23	22	nn0cnd 12544	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℂ)
24	14	neneqd 2962	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ 𝑦 = 0)
25	24	intnand 492	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ¬ (𝑥 = 0 ∧ 𝑦 = 0))
26		gcdeq0 16551	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
27	26	necon3abid 2993	. . . . . . . 8 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) ≠ 0 ↔ ¬ (𝑥 = 0 ∧ 𝑦 = 0)))
28	27	biimpar 481	. . . . . . 7 ⊢ (((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ¬ (𝑥 = 0 ∧ 𝑦 = 0)) → (𝑥 gcd 𝑦) ≠ 0)
29	11, 13, 25, 28	syl21anc 848	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ≠ 0)
30	20, 21, 23, 14, 29	divcan7d 11995	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) = (𝑥 / 𝑦))
31	19, 30	eqtr4d 2800	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
32		divgcdcoprm0 16699	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
33	11, 13, 14, 32	syl3anc 1390	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
34	31, 33	jca 519	. . 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 3595	. 2 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))
36		elq 12951	. . 3 ⊢ (𝑄 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
37	36	biimpi 218	. 2 ⊢ (𝑄 ∈ ℚ → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝑄 = (𝑥 / 𝑦))
38	35, 37	r19.29vva 3222	1 ⊢ (𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∃wrex 3086  (class class class)co 7396  0cc0 11073  1c1 11074   / cdiv 11844  ℕcn 12210  ℤcz 12568  ℚcq 12949   gcd cgcd 16528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-cnex 11129  ax-resscn 11130  ax-1cn 11131  ax-icn 11132  ax-addcl 11133  ax-addrcl 11134  ax-mulcl 11135  ax-mulrcl 11136  ax-mulcom 11137  ax-addass 11138  ax-mulass 11139  ax-distr 11140  ax-i2m1 11141  ax-1ne0 11142  ax-1rid 11143  ax-rnegex 11144  ax-rrecex 11145  ax-cnre 11146  ax-pre-lttri 11147  ax-pre-lttrn 11148  ax-pre-ltadd 11149  ax-pre-mulgt0 11150  ax-pre-sup 11151

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-sup 9388  df-inf 9389  df-pnf 11218  df-mnf 11219  df-xr 11220  df-ltxr 11221  df-le 11222  df-sub 11416  df-neg 11417  df-div 11845  df-nn 12211  df-2 12280  df-3 12281  df-n0 12482  df-z 12569  df-uz 12840  df-q 12950  df-rp 12994  df-fl 13802  df-mod 13880  df-seq 14015  df-exp 14075  df-cj 15126  df-re 15127  df-im 15128  df-sqrt 15262  df-abs 15263  df-dvds 16287  df-gcd 16529

This theorem is referenced by:  cos9thpiminplylem2  34080