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Theorem pellexlem3 15773

Description: Lemma for pellex . To each good rational approximation of (√‘𝐷), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)

**Assertion**
Ref	Expression
pellexlem3	⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})

Distinct variable group: 𝑥,𝐷,𝑦,𝑧

Proof of Theorem pellexlem3 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnex 9192	. . . 4 ⊢ ℕ ∈ V
2	1, 1	xpex 4848	. . 3 ⊢ (ℕ × ℕ) ∈ V
3		opabssxp 4806	. . 3 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ⊆ (ℕ × ℕ)
4	2, 3	ssexi 4232	. 2 ⊢ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∈ V
5		breq2 4097	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (0 < 𝑥 ↔ 0 < 𝑎))
6		fvoveq1 6051	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (abs‘(𝑥 − (√‘𝐷))) = (abs‘(𝑎 − (√‘𝐷))))
7		fveq2 5648	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (denom‘𝑥) = (denom‘𝑎))
8	7	oveq1d 6043	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((denom‘𝑥)↑-2) = ((denom‘𝑎)↑-2))
9	6, 8	breq12d 4106	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ((abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2) ↔ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))
10	5, 9	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝑎 → ((0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2)) ↔ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2))))
11	10	elrab 2963	. . . . . 6 ⊢ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ↔ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2))))
12		simprl 531	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → 𝑎 ∈ ℚ)
13		simprrl 541	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → 0 < 𝑎)
14		qgt0numnn 12832	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℚ ∧ 0 < 𝑎) → (numer‘𝑎) ∈ ℕ)
15	12, 13, 14	syl2anc 411	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (numer‘𝑎) ∈ ℕ)
16		qdencl 12822	. . . . . . . . 9 ⊢ (𝑎 ∈ ℚ → (denom‘𝑎) ∈ ℕ)
17	12, 16	syl 14	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (denom‘𝑎) ∈ ℕ)
18	15, 17	jca 306	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → ((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ))
19		simpll 527	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → 𝐷 ∈ ℕ)
20		simplr 529	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → ¬ (√‘𝐷) ∈ ℚ)
21		pellexlem1 15771	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ (numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ¬ (√‘𝐷) ∈ ℚ) → (((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0)
22	19, 15, 17, 20, 21	syl31anc 1277	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0)
23		simprrr 542	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2))
24		qeqnumdivden 12827	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ℚ → 𝑎 = ((numer‘𝑎) / (denom‘𝑎)))
25	24	oveq1d 6043	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ ℚ → (𝑎 − (√‘𝐷)) = (((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷)))
26	25	fveq2d 5652	. . . . . . . . . . 11 ⊢ (𝑎 ∈ ℚ → (abs‘(𝑎 − (√‘𝐷))) = (abs‘(((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷))))
27	26	breq1d 4103	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℚ → ((abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2) ↔ (abs‘(((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷))) < ((denom‘𝑎)↑-2)))
28	12, 27	syl 14	. . . . . . . . 9 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → ((abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2) ↔ (abs‘(((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷))) < ((denom‘𝑎)↑-2)))
29	23, 28	mpbid 147	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (abs‘(((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷))) < ((denom‘𝑎)↑-2))
30		pellexlem2 15772	. . . . . . . 8 ⊢ (((𝐷 ∈ ℕ ∧ (numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ (abs‘(((numer‘𝑎) / (denom‘𝑎)) − (√‘𝐷))) < ((denom‘𝑎)↑-2)) → (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷))))
31	19, 15, 17, 29, 30	syl31anc 1277	. . . . . . 7 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷))))
32	18, 22, 31	jca32 310	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷))))))
33	11, 32	sylan2b 287	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))}) → (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷))))))
34		qnumcl 12821	. . . . . . . 8 ⊢ (𝑎 ∈ ℚ → (numer‘𝑎) ∈ ℤ)
35		eleq1 2294	. . . . . . . . . . 11 ⊢ (𝑦 = (numer‘𝑎) → (𝑦 ∈ ℕ ↔ (numer‘𝑎) ∈ ℕ))
36	35	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑦 = (numer‘𝑎) → ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ↔ ((numer‘𝑎) ∈ ℕ ∧ 𝑧 ∈ ℕ)))
37		oveq1 6035	. . . . . . . . . . . . 13 ⊢ (𝑦 = (numer‘𝑎) → (𝑦↑2) = ((numer‘𝑎)↑2))
38	37	oveq1d 6043	. . . . . . . . . . . 12 ⊢ (𝑦 = (numer‘𝑎) → ((𝑦↑2) − (𝐷 · (𝑧↑2))) = (((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))))
39	38	neeq1d 2421	. . . . . . . . . . 11 ⊢ (𝑦 = (numer‘𝑎) → (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ↔ (((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0))
40	38	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑦 = (numer‘𝑎) → (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) = (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))))
41	40	breq1d 4103	. . . . . . . . . . 11 ⊢ (𝑦 = (numer‘𝑎) → ((abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))) ↔ (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))
42	39, 41	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑦 = (numer‘𝑎) → ((((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))) ↔ ((((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))))
43	36, 42	anbi12d 473	. . . . . . . . 9 ⊢ (𝑦 = (numer‘𝑎) → (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) ↔ (((numer‘𝑎) ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))))
44		eleq1 2294	. . . . . . . . . . 11 ⊢ (𝑧 = (denom‘𝑎) → (𝑧 ∈ ℕ ↔ (denom‘𝑎) ∈ ℕ))
45	44	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑧 = (denom‘𝑎) → (((numer‘𝑎) ∈ ℕ ∧ 𝑧 ∈ ℕ) ↔ ((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ)))
46		oveq1 6035	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (denom‘𝑎) → (𝑧↑2) = ((denom‘𝑎)↑2))
47	46	oveq2d 6044	. . . . . . . . . . . . 13 ⊢ (𝑧 = (denom‘𝑎) → (𝐷 · (𝑧↑2)) = (𝐷 · ((denom‘𝑎)↑2)))
48	47	oveq2d 6044	. . . . . . . . . . . 12 ⊢ (𝑧 = (denom‘𝑎) → (((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) = (((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))))
49	48	neeq1d 2421	. . . . . . . . . . 11 ⊢ (𝑧 = (denom‘𝑎) → ((((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ↔ (((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0))
50	48	fveq2d 5652	. . . . . . . . . . . 12 ⊢ (𝑧 = (denom‘𝑎) → (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) = (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))))
51	50	breq1d 4103	. . . . . . . . . . 11 ⊢ (𝑧 = (denom‘𝑎) → ((abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))) ↔ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))
52	49, 51	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑧 = (denom‘𝑎) → (((((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))) ↔ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷))))))
53	45, 52	anbi12d 473	. . . . . . . . 9 ⊢ (𝑧 = (denom‘𝑎) → ((((numer‘𝑎) ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷))))) ↔ (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))))
54	43, 53	opelopabg 4368	. . . . . . . 8 ⊢ (((numer‘𝑎) ∈ ℤ ∧ (denom‘𝑎) ∈ ℕ) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))))
55	34, 16, 54	syl2anc 411	. . . . . . 7 ⊢ (𝑎 ∈ ℚ → (⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))))
56	12, 55	syl 14	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ ℚ ∧ (0 < 𝑎 ∧ (abs‘(𝑎 − (√‘𝐷))) < ((denom‘𝑎)↑-2)))) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))))
57	11, 56	sylan2b 287	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))}) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ↔ (((numer‘𝑎) ∈ ℕ ∧ (denom‘𝑎) ∈ ℕ) ∧ ((((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2))) ≠ 0 ∧ (abs‘(((numer‘𝑎)↑2) − (𝐷 · ((denom‘𝑎)↑2)))) < (1 + (2 · (√‘𝐷)))))))
58	33, 57	mpbird 167	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ 𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))}) → ⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})
59	58	ex 115	. . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} → ⟨(numer‘𝑎), (denom‘𝑎)⟩ ∈ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}))
60		ssrab2 3313	. . . . . 6 ⊢ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ⊆ ℚ
61		simprl 531	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})) → 𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})
62	60, 61	sselid 3226	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})) → 𝑎 ∈ ℚ)
63		simprr 533	. . . . . 6 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})) → 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})
64	60, 63	sselid 3226	. . . . 5 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})) → 𝑏 ∈ ℚ)
65		opthg 4336	. . . . . . . . 9 ⊢ (((numer‘𝑎) ∈ ℤ ∧ (denom‘𝑎) ∈ ℕ) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))))
66	34, 16, 65	syl2anc 411	. . . . . . . 8 ⊢ (𝑎 ∈ ℚ → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))))
67	66	adantr 276	. . . . . . 7 ⊢ ((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))))
68		simprl 531	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → (numer‘𝑎) = (numer‘𝑏))
69		simprr 533	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → (denom‘𝑎) = (denom‘𝑏))
70	68, 69	oveq12d 6046	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → ((numer‘𝑎) / (denom‘𝑎)) = ((numer‘𝑏) / (denom‘𝑏)))
71		simpll 527	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → 𝑎 ∈ ℚ)
72	71, 24	syl 14	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → 𝑎 = ((numer‘𝑎) / (denom‘𝑎)))
73		simplr 529	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → 𝑏 ∈ ℚ)
74		qeqnumdivden 12827	. . . . . . . . . 10 ⊢ (𝑏 ∈ ℚ → 𝑏 = ((numer‘𝑏) / (denom‘𝑏)))
75	73, 74	syl 14	. . . . . . . . 9 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → 𝑏 = ((numer‘𝑏) / (denom‘𝑏)))
76	70, 72, 75	3eqtr4d 2274	. . . . . . . 8 ⊢ (((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) ∧ ((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏))) → 𝑎 = 𝑏)
77	76	ex 115	. . . . . . 7 ⊢ ((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) → (((numer‘𝑎) = (numer‘𝑏) ∧ (denom‘𝑎) = (denom‘𝑏)) → 𝑎 = 𝑏))
78	67, 77	sylbid 150	. . . . . 6 ⊢ ((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ → 𝑎 = 𝑏))
79		fveq2 5648	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (numer‘𝑎) = (numer‘𝑏))
80		fveq2 5648	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (denom‘𝑎) = (denom‘𝑏))
81	79, 80	opeq12d 3875	. . . . . 6 ⊢ (𝑎 = 𝑏 → ⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩)
82	78, 81	impbid1 142	. . . . 5 ⊢ ((𝑎 ∈ ℚ ∧ 𝑏 ∈ ℚ) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ 𝑎 = 𝑏))
83	62, 64, 82	syl2anc 411	. . . 4 ⊢ (((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) ∧ (𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))})) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ 𝑎 = 𝑏))
84	83	ex 115	. . 3 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝑎 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ∧ 𝑏 ∈ {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))}) → (⟨(numer‘𝑎), (denom‘𝑎)⟩ = ⟨(numer‘𝑏), (denom‘𝑏)⟩ ↔ 𝑎 = 𝑏)))
85	59, 84	dom2d 6989	. 2 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ({⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ∈ V → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))}))
86	4, 85	mpi 15	1 ⊢ ((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1398 ∈ wcel 2202 ≠ wne 2403 {crab 2515 Vcvv 2803 ⟨cop 3676 class class class wbr 4093 {copab 4154 × cxp 4729 ‘cfv 5333 (class class class)co 6028 ≼ cdom 6951 0cc0 8075 1c1 8076 + caddc 8078 · cmul 8080 < clt 8257 − cmin 8393 -cneg 8394 / cdiv 8895 ℕcn 9186 2c2 9237 ℤcz 9522 ℚcq 9896 ↑cexp 10844 √csqrt 11617 abscabs 11618 numercnumer 12814 denomcdenom 12815

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-dom 6954 df-sup 7226 df-pnf 8259 df-mnf 8260 df-xr 8261 df-ltxr 8262 df-le 8263 df-sub 8395 df-neg 8396 df-reap 8798 df-ap 8805 df-div 8896 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-n0 9446 df-z 9523 df-uz 9799 df-q 9897 df-rp 9932 df-fz 10287 df-fzo 10421 df-fl 10574 df-mod 10629 df-seqfrec 10754 df-exp 10845 df-cj 11463 df-re 11464 df-im 11465 df-rsqrt 11619 df-abs 11620 df-dvds 12410 df-gcd 12586 df-numer 12816 df-denom 12817

This theorem is referenced by: (None)

W3C validator