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Theorem divalglemeunn 12688

Description: Lemma for divalg 12691. Uniqueness for a positive denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)

**Assertion**
Ref	Expression
divalglemeunn	⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

Distinct variable groups: 𝐷,𝑞,𝑟 𝑁,𝑞,𝑟

Proof of Theorem divalglemeunn Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		divalglemnn 12685	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
2		nfv 1581	. . . . . 6 ⊢ Ⅎ𝑞((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ))
3		nfre1 2593	. . . . . . 7 ⊢ Ⅎ𝑞∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))
4		nfv 1581	. . . . . . 7 ⊢ Ⅎ𝑞 𝑟 = 𝑠
5	3, 4	nfim 1625	. . . . . 6 ⊢ Ⅎ𝑞(∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠)
6		oveq1 6092	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑡 → (𝑞 · 𝐷) = (𝑡 · 𝐷))
7	6	oveq1d 6100	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → ((𝑞 · 𝐷) + 𝑠) = ((𝑡 · 𝐷) + 𝑠))
8	7	eqeq2d 2250	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (𝑁 = ((𝑞 · 𝐷) + 𝑠) ↔ 𝑁 = ((𝑡 · 𝐷) + 𝑠)))
9	8	3anbi3d 1359	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → ((0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))))
10	9	cbvrexv 2787	. . . . . . . 8 ⊢ (∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) ↔ ∃𝑡 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠)))
11		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 < 𝑡) → 𝑞 < 𝑡)
12		simplr 533	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) → 𝐷 ∈ ℕ)
13	12	ad4antr 498	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝐷 ∈ ℕ)
14		simplrl 541	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑟 ∈ ℤ)
15	14	ad3antrrr 496	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑟 ∈ ℤ)
16		simplrr 542	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑠 ∈ ℤ)
17	16	ad3antrrr 496	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑠 ∈ ℤ)
18		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℤ)
19	18	ad3antrrr 496	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑞 ∈ ℤ)
20		simplr 533	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑡 ∈ ℤ)
21		simpr1 1034	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 0 ≤ 𝑠)
22		simpr2 1035	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 𝑟 < (abs‘𝐷))
23	22	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑟 < (abs‘𝐷))
24	13	nnred 9317	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝐷 ∈ ℝ)
25	13	nnnn0d 9620	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝐷 ∈ ℕ₀)
26	25	nn0ge0d 9623	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 0 ≤ 𝐷)
27	24, 26	absidd 11933	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → (abs‘𝐷) = 𝐷)
28	23, 27	breqtrd 4156	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑟 < 𝐷)
29		simpr3 1036	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 𝑁 = ((𝑞 · 𝐷) + 𝑟))
30	29	ad2antrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑁 = ((𝑞 · 𝐷) + 𝑟))
31		simpr3 1036	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑁 = ((𝑡 · 𝐷) + 𝑠))
32	30, 31	eqtr3d 2273	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → ((𝑞 · 𝐷) + 𝑟) = ((𝑡 · 𝐷) + 𝑠))
33	13, 15, 17, 19, 20, 21, 28, 32	divalglemnqt 12687	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → ¬ 𝑞 < 𝑡)
34	33	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 < 𝑡) → ¬ 𝑞 < 𝑡)
35	11, 34	pm2.21dd 629	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 < 𝑡) → 𝑟 = 𝑠)
36	13	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝐷 ∈ ℕ)
37	36	nnzd 9767	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝐷 ∈ ℤ)
38	15	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑟 ∈ ℤ)
39	17	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑠 ∈ ℤ)
40	19	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑞 ∈ ℤ)
41	20	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑡 ∈ ℤ)
42		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑞 = 𝑡)
43	32	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → ((𝑞 · 𝐷) + 𝑟) = ((𝑡 · 𝐷) + 𝑠))
44	37, 38, 39, 40, 41, 42, 43	divalglemqt 12686	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑞 = 𝑡) → 𝑟 = 𝑠)
45		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑡 < 𝑞) → 𝑡 < 𝑞)
46		simpr1 1034	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 0 ≤ 𝑟)
47	46	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 0 ≤ 𝑟)
48		simpr2 1035	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑠 < (abs‘𝐷))
49	48, 27	breqtrd 4156	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑠 < 𝐷)
50	31, 30	eqtr3d 2273	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → ((𝑡 · 𝐷) + 𝑠) = ((𝑞 · 𝐷) + 𝑟))
51	13, 17, 15, 20, 19, 47, 49, 50	divalglemnqt 12687	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → ¬ 𝑡 < 𝑞)
52	51	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑡 < 𝑞) → ¬ 𝑡 < 𝑞)
53	45, 52	pm2.21dd 629	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) ∧ 𝑡 < 𝑞) → 𝑟 = 𝑠)
54		ztri3or 9687	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑞 < 𝑡 ∨ 𝑞 = 𝑡 ∨ 𝑡 < 𝑞))
55	19, 20, 54	syl2anc 415	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → (𝑞 < 𝑡 ∨ 𝑞 = 𝑡 ∨ 𝑡 < 𝑞))
56	35, 44, 53, 55	mpjao3dan 1348	. . . . . . . . . 10 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) ∧ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))) → 𝑟 = 𝑠)
57	56	ex 115	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) ∧ 𝑡 ∈ ℤ) → ((0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠)) → 𝑟 = 𝑠))
58	57	rexlimdva 2668	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → (∃𝑡 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠)) → 𝑟 = 𝑠))
59	10, 58	biimtrid 152	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → (∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠))
60	59	exp31 364	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) → (𝑞 ∈ ℤ → ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) → (∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠))))
61	2, 5, 60	rexlimd 2665	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) → (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) → (∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠)))
62	61	impd 254	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) → ((∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ∧ ∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))) → 𝑟 = 𝑠))
63	62	ralrimivva 2632	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∀𝑟 ∈ ℤ ∀𝑠 ∈ ℤ ((∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ∧ ∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))) → 𝑟 = 𝑠))
64		breq2 4134	. . . . . 6 ⊢ (𝑟 = 𝑠 → (0 ≤ 𝑟 ↔ 0 ≤ 𝑠))
65		breq1 4133	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 < (abs‘𝐷) ↔ 𝑠 < (abs‘𝐷)))
66		oveq2 6093	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝑞 · 𝐷) + 𝑟) = ((𝑞 · 𝐷) + 𝑠))
67	66	eqeq2d 2250	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑁 = ((𝑞 · 𝐷) + 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑠)))
68	64, 65, 67	3anbi123d 1353	. . . . 5 ⊢ (𝑟 = 𝑠 → ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))))
69	68	rexbidv 2551	. . . 4 ⊢ (𝑟 = 𝑠 → (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))))
70	69	rmo4 3019	. . 3 ⊢ (∃*𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∀𝑟 ∈ ℤ ∀𝑠 ∈ ℤ ((∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ∧ ∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))) → 𝑟 = 𝑠))
71	63, 70	sylibr 134	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃*𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
72		reu5 2770	. 2 ⊢ (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ∧ ∃*𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))))
73	1, 71, 72	sylanbrc 421	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℕ) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ∨ w3o 1008 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 ∃!wreu 2530 ∃*wrmo 2531 class class class wbr 4130 ‘cfv 5377 (class class class)co 6085 0cc0 8179 + caddc 8182 · cmul 8184 < clt 8360 ≤ cle 8361 ℕcn 9304 ℤcz 9644 abscabs 11763

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-n0 9564 df-z 9645 df-uz 9922 df-q 10020 df-rp 10055 df-fl 10705 df-mod 10760 df-seqfrec 10885 df-exp 10976 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765

This theorem is used by: divalg 12691

W3C validator