qdiff - Mathbox for Jim Kingdon

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Theorem qdiff 37661

Description: The rationals are exactly those reals for which there exist two distinct rationals that are the same distance from the original number. Similar to irrdiff 37660 but here proved with a proof which would also work in constructive mathematics. From an online post by Ingo Blechschmidt. For a proof using irrdiff 37660, see qdiffALT 37662. (Contributed by Jim Kingdon, 24-Apr-2026.)

**Assertion**
Ref	Expression
qdiff	⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))

Distinct variable group: 𝐴,𝑞,𝑟

**Proof of Theorem qdiff**
Step	Hyp	Ref	Expression
1		neeq1 2995	. . . . 5 ⊢ (𝑞 = (𝐴 − 1) → (𝑞 ≠ 𝑟 ↔ (𝐴 − 1) ≠ 𝑟))
2		oveq2 7370	. . . . . 6 ⊢ (𝑞 = (𝐴 − 1) → (𝐴 − 𝑞) = (𝐴 − (𝐴 − 1)))
3	2	fveqeq2d 6844	. . . . 5 ⊢ (𝑞 = (𝐴 − 1) → ((abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)) ↔ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟))))
4	1, 3	anbi12d 633	. . . 4 ⊢ (𝑞 = (𝐴 − 1) → ((𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))) ↔ ((𝐴 − 1) ≠ 𝑟 ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟)))))
5	4	rexbidv 3162	. . 3 ⊢ (𝑞 = (𝐴 − 1) → (∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))) ↔ ∃𝑟 ∈ ℚ ((𝐴 − 1) ≠ 𝑟 ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟)))))
6		1z 12552	. . . . 5 ⊢ 1 ∈ ℤ
7		zq 12899	. . . . 5 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
8	6, 7	ax-mp 5	. . . 4 ⊢ 1 ∈ ℚ
9		qsubcl 12913	. . . 4 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 − 1) ∈ ℚ)
10	8, 9	mpan2 692	. . 3 ⊢ (𝐴 ∈ ℚ → (𝐴 − 1) ∈ ℚ)
11		qaddcl 12910	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 1 ∈ ℚ) → (𝐴 + 1) ∈ ℚ)
12	8, 11	mpan2 692	. . . 4 ⊢ (𝐴 ∈ ℚ → (𝐴 + 1) ∈ ℚ)
13		qre 12898	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
14		1rp 12941	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℝ⁺
15	14, 14	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺)
16		rpaddcl 12961	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ⁺ ∧ 1 ∈ ℝ⁺) → (1 + 1) ∈ ℝ⁺)
17	15, 16	mp1i 13	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℚ → (1 + 1) ∈ ℝ⁺)
18	13, 17	ltaddrpd 13014	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 < (𝐴 + (1 + 1)))
19	13, 18	ltned 11277	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → 𝐴 ≠ (𝐴 + (1 + 1)))
20	19	neneqd 2938	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → ¬ 𝐴 = (𝐴 + (1 + 1)))
21	20	neqcomd 2747	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ¬ (𝐴 + (1 + 1)) = 𝐴)
22		qcn 12908	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
23		1cnd 11134	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → 1 ∈ ℂ)
24	22, 23, 23	addassd 11162	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → ((𝐴 + 1) + 1) = (𝐴 + (1 + 1)))
25	24	eqeq1d 2739	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (((𝐴 + 1) + 1) = 𝐴 ↔ (𝐴 + (1 + 1)) = 𝐴))
26	21, 25	mtbird 325	. . . . . 6 ⊢ (𝐴 ∈ ℚ → ¬ ((𝐴 + 1) + 1) = 𝐴)
27	22, 23	addcld 11159	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (𝐴 + 1) ∈ ℂ)
28	22, 23, 27	subadd2d 11519	. . . . . 6 ⊢ (𝐴 ∈ ℚ → ((𝐴 − 1) = (𝐴 + 1) ↔ ((𝐴 + 1) + 1) = 𝐴))
29	26, 28	mtbird 325	. . . . 5 ⊢ (𝐴 ∈ ℚ → ¬ (𝐴 − 1) = (𝐴 + 1))
30	29	neqned 2940	. . . 4 ⊢ (𝐴 ∈ ℚ → (𝐴 − 1) ≠ (𝐴 + 1))
31	23	absnegd 15409	. . . . 5 ⊢ (𝐴 ∈ ℚ → (abs‘-1) = (abs‘1))
32	22, 22, 23	subsub4d 11531	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ((𝐴 − 𝐴) − 1) = (𝐴 − (𝐴 + 1)))
33	22	subidd 11488	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (𝐴 − 𝐴) = 0)
34	33	oveq1d 7377	. . . . . . . 8 ⊢ (𝐴 ∈ ℚ → ((𝐴 − 𝐴) − 1) = (0 − 1))
35		df-neg 11375	. . . . . . . 8 ⊢ -1 = (0 − 1)
36	34, 35	eqtr4di 2790	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → ((𝐴 − 𝐴) − 1) = -1)
37	32, 36	eqtr3d 2774	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (𝐴 − (𝐴 + 1)) = -1)
38	37	fveq2d 6840	. . . . 5 ⊢ (𝐴 ∈ ℚ → (abs‘(𝐴 − (𝐴 + 1))) = (abs‘-1))
39	22, 23	nncand 11505	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (𝐴 − (𝐴 − 1)) = 1)
40	39	fveq2d 6840	. . . . 5 ⊢ (𝐴 ∈ ℚ → (abs‘(𝐴 − (𝐴 − 1))) = (abs‘1))
41	31, 38, 40	3eqtr4rd 2783	. . . 4 ⊢ (𝐴 ∈ ℚ → (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − (𝐴 + 1))))
42		neeq2 2996	. . . . . 6 ⊢ (𝑟 = (𝐴 + 1) → ((𝐴 − 1) ≠ 𝑟 ↔ (𝐴 − 1) ≠ (𝐴 + 1)))
43		oveq2 7370	. . . . . . . 8 ⊢ (𝑟 = (𝐴 + 1) → (𝐴 − 𝑟) = (𝐴 − (𝐴 + 1)))
44	43	fveq2d 6840	. . . . . . 7 ⊢ (𝑟 = (𝐴 + 1) → (abs‘(𝐴 − 𝑟)) = (abs‘(𝐴 − (𝐴 + 1))))
45	44	eqeq2d 2748	. . . . . 6 ⊢ (𝑟 = (𝐴 + 1) → ((abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟)) ↔ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − (𝐴 + 1)))))
46	42, 45	anbi12d 633	. . . . 5 ⊢ (𝑟 = (𝐴 + 1) → (((𝐴 − 1) ≠ 𝑟 ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟))) ↔ ((𝐴 − 1) ≠ (𝐴 + 1) ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − (𝐴 + 1))))))
47	46	rspcev 3565	. . . 4 ⊢ (((𝐴 + 1) ∈ ℚ ∧ ((𝐴 − 1) ≠ (𝐴 + 1) ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − (𝐴 + 1))))) → ∃𝑟 ∈ ℚ ((𝐴 − 1) ≠ 𝑟 ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟))))
48	12, 30, 41, 47	syl12anc 837	. . 3 ⊢ (𝐴 ∈ ℚ → ∃𝑟 ∈ ℚ ((𝐴 − 1) ≠ 𝑟 ∧ (abs‘(𝐴 − (𝐴 − 1))) = (abs‘(𝐴 − 𝑟))))
49	5, 10, 48	rspcedvdw 3568	. 2 ⊢ (𝐴 ∈ ℚ → ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))))
50		2cnd 12254	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 2 ∈ ℂ)
51		simpll 767	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝐴 ∈ ℝ)
52	51	recnd 11168	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝐴 ∈ ℂ)
53		simplrl 777	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑞 ∈ ℚ)
54	53	qred 12900	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑞 ∈ ℝ)
55	54	recnd 11168	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑞 ∈ ℂ)
56	52, 55	mulcld 11160	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴 · 𝑞) ∈ ℂ)
57		simplrr 778	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑟 ∈ ℚ)
58	57	qred 12900	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑟 ∈ ℝ)
59	58	recnd 11168	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑟 ∈ ℂ)
60	52, 59	mulcld 11160	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴 · 𝑟) ∈ ℂ)
61	50, 56, 60	subdid 11601	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (2 · ((𝐴 · 𝑞) − (𝐴 · 𝑟))) = ((2 · (𝐴 · 𝑞)) − (2 · (𝐴 · 𝑟))))
62	52	sqcld 14101	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴↑2) ∈ ℂ)
63	50, 60	mulcld 11160	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (2 · (𝐴 · 𝑟)) ∈ ℂ)
64	50, 56	mulcld 11160	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (2 · (𝐴 · 𝑞)) ∈ ℂ)
65	62, 63, 64	nnncan1d 11534	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝐴↑2) − (2 · (𝐴 · 𝑟))) − ((𝐴↑2) − (2 · (𝐴 · 𝑞)))) = ((2 · (𝐴 · 𝑞)) − (2 · (𝐴 · 𝑟))))
66		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))
67	51, 54	resubcld 11573	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴 − 𝑞) ∈ ℝ)
68	51, 58	resubcld 11573	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴 − 𝑟) ∈ ℝ)
69		sqabs 15264	. . . . . . . . . . . . 13 ⊢ (((𝐴 − 𝑞) ∈ ℝ ∧ (𝐴 − 𝑟) ∈ ℝ) → (((𝐴 − 𝑞)↑2) = ((𝐴 − 𝑟)↑2) ↔ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))))
70	67, 68, 69	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝐴 − 𝑞)↑2) = ((𝐴 − 𝑟)↑2) ↔ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))))
71	66, 70	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴 − 𝑞)↑2) = ((𝐴 − 𝑟)↑2))
72		binom2sub 14177	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑞 ∈ ℂ) → ((𝐴 − 𝑞)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝑞))) + (𝑞↑2)))
73	52, 55, 72	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴 − 𝑞)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝑞))) + (𝑞↑2)))
74		binom2sub 14177	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℂ ∧ 𝑟 ∈ ℂ) → ((𝐴 − 𝑟)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝑟))) + (𝑟↑2)))
75	52, 59, 74	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴 − 𝑟)↑2) = (((𝐴↑2) − (2 · (𝐴 · 𝑟))) + (𝑟↑2)))
76	71, 73, 75	3eqtr3d 2780	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝐴↑2) − (2 · (𝐴 · 𝑞))) + (𝑞↑2)) = (((𝐴↑2) − (2 · (𝐴 · 𝑟))) + (𝑟↑2)))
77	62, 64	subcld 11500	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴↑2) − (2 · (𝐴 · 𝑞))) ∈ ℂ)
78	55	sqcld 14101	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑞↑2) ∈ ℂ)
79	62, 63	subcld 11500	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴↑2) − (2 · (𝐴 · 𝑟))) ∈ ℂ)
80	59	sqcld 14101	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑟↑2) ∈ ℂ)
81	77, 78, 79, 80	addsubeq4d 11551	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((((𝐴↑2) − (2 · (𝐴 · 𝑞))) + (𝑞↑2)) = (((𝐴↑2) − (2 · (𝐴 · 𝑟))) + (𝑟↑2)) ↔ (((𝐴↑2) − (2 · (𝐴 · 𝑟))) − ((𝐴↑2) − (2 · (𝐴 · 𝑞)))) = ((𝑞↑2) − (𝑟↑2))))
82	76, 81	mpbid 232	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝐴↑2) − (2 · (𝐴 · 𝑟))) − ((𝐴↑2) − (2 · (𝐴 · 𝑞)))) = ((𝑞↑2) − (𝑟↑2)))
83	61, 65, 82	3eqtr2d 2778	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (2 · ((𝐴 · 𝑞) − (𝐴 · 𝑟))) = ((𝑞↑2) − (𝑟↑2)))
84	78, 80	subcld 11500	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝑞↑2) − (𝑟↑2)) ∈ ℂ)
85	56, 60	subcld 11500	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝐴 · 𝑞) − (𝐴 · 𝑟)) ∈ ℂ)
86		2ne0 12280	. . . . . . . . . 10 ⊢ 2 ≠ 0
87	86	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 2 ≠ 0)
88	84, 50, 85, 87	divmuld 11948	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((((𝑞↑2) − (𝑟↑2)) / 2) = ((𝐴 · 𝑞) − (𝐴 · 𝑟)) ↔ (2 · ((𝐴 · 𝑞) − (𝐴 · 𝑟))) = ((𝑞↑2) − (𝑟↑2))))
89	83, 88	mpbird 257	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝑞↑2) − (𝑟↑2)) / 2) = ((𝐴 · 𝑞) − (𝐴 · 𝑟)))
90	52, 55, 59	subdid 11601	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝐴 · (𝑞 − 𝑟)) = ((𝐴 · 𝑞) − (𝐴 · 𝑟)))
91	89, 90	eqtr4d 2775	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝑞↑2) − (𝑟↑2)) / 2) = (𝐴 · (𝑞 − 𝑟)))
92	84	halfcld 12417	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝑞↑2) − (𝑟↑2)) / 2) ∈ ℂ)
93	55, 59	subcld 11500	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑞 − 𝑟) ∈ ℂ)
94		simprl 771	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝑞 ≠ 𝑟)
95	55, 59, 94	subne0d 11509	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑞 − 𝑟) ≠ 0)
96	92, 52, 93, 95	divmul3d 11960	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((((𝑞↑2) − (𝑟↑2)) / 2) / (𝑞 − 𝑟)) = 𝐴 ↔ (((𝑞↑2) − (𝑟↑2)) / 2) = (𝐴 · (𝑞 − 𝑟))))
97	91, 96	mpbird 257	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((((𝑞↑2) − (𝑟↑2)) / 2) / (𝑞 − 𝑟)) = 𝐴)
98		qsqcl 14087	. . . . . . . . 9 ⊢ (𝑞 ∈ ℚ → (𝑞↑2) ∈ ℚ)
99	53, 98	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑞↑2) ∈ ℚ)
100		qsqcl 14087	. . . . . . . . 9 ⊢ (𝑟 ∈ ℚ → (𝑟↑2) ∈ ℚ)
101	57, 100	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑟↑2) ∈ ℚ)
102		qsubcl 12913	. . . . . . . 8 ⊢ (((𝑞↑2) ∈ ℚ ∧ (𝑟↑2) ∈ ℚ) → ((𝑞↑2) − (𝑟↑2)) ∈ ℚ)
103	99, 101, 102	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((𝑞↑2) − (𝑟↑2)) ∈ ℚ)
104		2z 12554	. . . . . . . 8 ⊢ 2 ∈ ℤ
105		zq 12899	. . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ ℚ)
106	104, 105	mp1i 13	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 2 ∈ ℚ)
107		qdivcl 12915	. . . . . . 7 ⊢ ((((𝑞↑2) − (𝑟↑2)) ∈ ℚ ∧ 2 ∈ ℚ ∧ 2 ≠ 0) → (((𝑞↑2) − (𝑟↑2)) / 2) ∈ ℚ)
108	103, 106, 87, 107	syl3anc 1374	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (((𝑞↑2) − (𝑟↑2)) / 2) ∈ ℚ)
109		qsubcl 12913	. . . . . . 7 ⊢ ((𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ) → (𝑞 − 𝑟) ∈ ℚ)
110	53, 57, 109	syl2anc 585	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → (𝑞 − 𝑟) ∈ ℚ)
111		qdivcl 12915	. . . . . 6 ⊢ (((((𝑞↑2) − (𝑟↑2)) / 2) ∈ ℚ ∧ (𝑞 − 𝑟) ∈ ℚ ∧ (𝑞 − 𝑟) ≠ 0) → ((((𝑞↑2) − (𝑟↑2)) / 2) / (𝑞 − 𝑟)) ∈ ℚ)
112	108, 110, 95, 111	syl3anc 1374	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → ((((𝑞↑2) − (𝑟↑2)) / 2) / (𝑞 − 𝑟)) ∈ ℚ)
113	97, 112	eqeltrrd 2838	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) ∧ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))) → 𝐴 ∈ ℚ)
114	113	ex 412	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑞 ∈ ℚ ∧ 𝑟 ∈ ℚ)) → ((𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))) → 𝐴 ∈ ℚ))
115	114	rexlimdvva 3195	. 2 ⊢ (𝐴 ∈ ℝ → (∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟))) → 𝐴 ∈ ℚ))
116	49, 115	impbid2 226	1 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℚ ↔ ∃𝑞 ∈ ℚ ∃𝑟 ∈ ℚ (𝑞 ≠ 𝑟 ∧ (abs‘(𝐴 − 𝑞)) = (abs‘(𝐴 − 𝑟)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 ‘cfv 6494 (class class class)co 7362 ℂcc 11031 ℝcr 11032 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 − cmin 11372 -cneg 11373 / cdiv 11802 2c2 12231 ℤcz 12519 ℚcq 12893 ℝ⁺crp 12937 ↑cexp 14018 abscabs 15191

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 5232 ax-nul 5242 ax-pow 5304 ax-pr 5372 ax-un 7684 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

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 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5521 df-eprel 5526 df-po 5534 df-so 5535 df-fr 5579 df-we 5581 df-xp 5632 df-rel 5633 df-cnv 5634 df-co 5635 df-dm 5636 df-rn 5637 df-res 5638 df-ima 5639 df-pred 6261 df-ord 6322 df-on 6323 df-lim 6324 df-suc 6325 df-iota 6450 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7319 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7813 df-1st 7937 df-2nd 7938 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-er 8638 df-en 8889 df-dom 8890 df-sdom 8891 df-sup 9350 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193

This theorem is referenced by: (None)

W3C validator