Theorem elq2 32903

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 7375	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞))
2	1	eqeq2d 2748	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑄 = (𝑝 / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞)))
3		oveq1 7375	. . . . 5 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → (𝑝 gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞))
4	3	eqeq1d 2739	. . . 4 ⊢ (𝑝 = (𝑥 / (𝑥 gcd 𝑦)) → ((𝑝 gcd 𝑞) = 1 ↔ ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = 1))
5	2, 4	anbi12d 633	. . 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 7376	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
7	6	eqeq2d 2748	. . . 4 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → (𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / 𝑞) ↔ 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦)))))
8		oveq2 7376	. . . . 5 ⊢ (𝑞 = (𝑦 / (𝑥 gcd 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) gcd 𝑞) = ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))))
9	8	eqeq1d 2739	. . . 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 633	. . 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 776	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℤ)
12		simplr 769	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℕ)
13	12	nnzd 12526	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℤ)
14	12	nnne0d 12207	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ≠ 0)
15		divgcdz 16450	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
16	11, 13, 14, 15	syl3anc 1374	. . 3 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 / (𝑥 gcd 𝑦)) ∈ ℤ)
17		divgcdnnr 16455	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ 𝑥 ∈ ℤ) → (𝑦 / (𝑥 gcd 𝑦)) ∈ ℕ)
18	12, 11, 17	syl2anc 585	. . 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 12609	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑥 ∈ ℂ)
21	12	nncnd 12173	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑦 ∈ ℂ)
22	11, 13	gcdcld 16447	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℕ0)
23	22	nn0cnd 12476	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ∈ ℂ)
24	14	neneqd 2938	. . . . . . . 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 16456	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑥 gcd 𝑦) = 0 ↔ (𝑥 = 0 ∧ 𝑦 = 0)))
27	26	necon3abid 2969	. . . . . . . 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 838	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → (𝑥 gcd 𝑦) ≠ 0)
30	20, 21, 23, 14, 29	divcan7d 11957	. . . . 5 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))) = (𝑥 / 𝑦))
31	19, 30	eqtr4d 2775	. . . 4 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → 𝑄 = ((𝑥 / (𝑥 gcd 𝑦)) / (𝑦 / (𝑥 gcd 𝑦))))
32		divgcdcoprm0 16604	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑦 ≠ 0) → ((𝑥 / (𝑥 gcd 𝑦)) gcd (𝑦 / (𝑥 gcd 𝑦))) = 1)
33	11, 13, 14, 32	syl3anc 1374	. . . 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 3592	. 2 ⊢ ((((𝑄 ∈ ℚ ∧ 𝑥 ∈ ℤ) ∧ 𝑦 ∈ ℕ) ∧ 𝑄 = (𝑥 / 𝑦)) → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))
36		elq 12875	. . 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 3198	1 ⊢ (𝑄 ∈ ℚ → ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ (𝑄 = (𝑝 / 𝑞) ∧ (𝑝 gcd 𝑞) = 1))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  (class class class)co 7368  0cc0 11038  1c1 11039   / cdiv 11806  ℕcn 12157  ℤcz 12500  ℚcq 12873   gcd cgcd 16433

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 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

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 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-sup 9357  df-inf 9358  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-dvds 16192  df-gcd 16434

This theorem is referenced by:  cos9thpiminplylem2  33961