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Theorem divalglemeuneg 12234

Description: Lemma for divalg 12235. Uniqueness for a negative denominator. (Contributed by Jim Kingdon, 4-Dec-2021.)

**Assertion**
Ref	Expression
divalglemeuneg	⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃!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 divalglemeuneg Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1002	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 < 0)
2	1	lt0ne0d 8586	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ≠ 0)
3		divalglemex 12233	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
4	2, 3	syld3an3 1295	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
5		nfv 1551	. . . . . 6 ⊢ Ⅎ𝑞((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ))
6		nfre1 2549	. . . . . . 7 ⊢ Ⅎ𝑞∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))
7		nfv 1551	. . . . . . 7 ⊢ Ⅎ𝑞 𝑟 = 𝑠
8	6, 7	nfim 1595	. . . . . 6 ⊢ Ⅎ𝑞(∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠)
9		oveq1 5951	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑡 → (𝑞 · 𝐷) = (𝑡 · 𝐷))
10	9	oveq1d 5959	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → ((𝑞 · 𝐷) + 𝑠) = ((𝑡 · 𝐷) + 𝑠))
11	10	eqeq2d 2217	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (𝑁 = ((𝑞 · 𝐷) + 𝑠) ↔ 𝑁 = ((𝑡 · 𝐷) + 𝑠)))
12	11	3anbi3d 1331	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → ((0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))))
13	12	cbvrexv 2739	. . . . . . . 8 ⊢ (∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) ↔ ∃𝑡 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠)))
14		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
15		simp2 1001	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ∈ ℤ)
16	15	znegcld 9497	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → -𝐷 ∈ ℤ)
17	15	zred 9495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ∈ ℝ)
18	17	lt0neg1d 8588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → (𝐷 < 0 ↔ 0 < -𝐷))
19	1, 18	mpbid 147	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 0 < -𝐷)
20		elnnz 9382	. . . . . . . . . . . . . . . . 17 ⊢ (-𝐷 ∈ ℕ ↔ (-𝐷 ∈ ℤ ∧ 0 < -𝐷))
21	16, 19, 20	sylanbrc 417	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → -𝐷 ∈ ℕ)
22	21	ad5antr 496	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
23		simplrr 536	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑠 ∈ ℤ)
24	23	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
25		simplrl 535	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑟 ∈ ℤ)
26	25	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
27		simplr 528	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
28	27	znegcld 9497	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
29		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℤ)
30	29	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
31	30	znegcld 9497	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
32		simpr1 1006	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 0 ≤ 𝑟)
33	32	ad2antrr 488	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
34		simpr2 1007	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
35		simpll2 1040	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝐷 ∈ ℤ)
36	35	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℤ)
37	36	zred 9495	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
38		0red 8073	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
39		simpll3 1041	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝐷 < 0)
40	39	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 < 0)
41	37, 38, 40	ltled 8191	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ≤ 0)
42	37, 41	absnidd 11471	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
43	34, 42	breqtrd 4070	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
44		simpr3 1008	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
45	27	zcnd 9496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℂ)
46	36	zcnd 9496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℂ)
47	45, 46	mul2negd 8485	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) = (𝑡 · 𝐷))
48	47	oveq1d 5959	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) = ((𝑡 · 𝐷) + 𝑠))
49	44, 48	eqtr4d 2241	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
50		simpr3 1008	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 𝑁 = ((𝑞 · 𝐷) + 𝑟))
51	50	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
52	30	zcnd 9496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℂ)
53	52, 46	mul2negd 8485	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) = (𝑞 · 𝐷))
54	53	oveq1d 5959	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) = ((𝑞 · 𝐷) + 𝑟))
55	51, 54	eqtr4d 2241	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
56	49, 55	eqtr3d 2240	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
57	22, 24, 26, 28, 31, 33, 43, 56	divalglemnqt 12231	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
58	30	zred 9495	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℝ)
59	27	zred 9495	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ∈ ℝ)
60	58, 59	ltnegd 8596	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 < -𝑞))
61	57, 60	mtbird 675	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
62	61	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
63	14, 62	pm2.21dd 621	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
64	36	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
65	26	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
66	24	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
67	30	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
68	27	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
69		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
70	51	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) → 𝑁 = ((𝑞 · 𝐷) + 𝑟))
71	44	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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) → 𝑁 = ((𝑡 · 𝐷) + 𝑠))
72	70, 71	eqtr3d 2240	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
73	64, 65, 66, 67, 68, 69, 72	divalglemqt 12230	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
74		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
75		simpr1 1006	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
76		simpr2 1007	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 𝑟 < (abs‘𝐷))
77	76	ad2antrr 488	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
78	77, 42	breqtrd 4070	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
79	55, 49	eqtr3d 2240	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
80	22, 26, 24, 31, 28, 75, 78, 79	divalglemnqt 12231	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
81	59, 58	ltnegd 8596	. . . . . . . . . . . . . 14 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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 ↔ -𝑞 < -𝑡))
82	80, 81	mtbird 675	. . . . . . . . . . . . 13 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
83	82	adantr 276	. . . . . . . . . . . 12 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
84	74, 83	pm2.21dd 621	. . . . . . . . . . 11 ⊢ ((((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
85		simplr 528	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) ∧ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟))) → 𝑞 ∈ ℤ)
86	85	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
87		ztri3or 9415	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑞 < 𝑡 ∨ 𝑞 = 𝑡 ∨ 𝑡 < 𝑞))
88	86, 27, 87	syl2anc 411	. . . . . . . . . . 11 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
89	63, 73, 84, 88	mpjao3dan 1320	. . . . . . . . . 10 ⊢ (((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)
90	89	ex 115	. . . . . . . . 9 ⊢ ((((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
91	90	rexlimdva 2623	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
92	13, 91	biimtrid 152	. . . . . . 7 ⊢ (((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
93	92	exp31 364	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))))
94	5, 8, 93	rexlimd 2620	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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)))
95	94	impd 254	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (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))
96	95	ralrimivva 2588	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∀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))
97		breq2 4048	. . . . . 6 ⊢ (𝑟 = 𝑠 → (0 ≤ 𝑟 ↔ 0 ≤ 𝑠))
98		breq1 4047	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 < (abs‘𝐷) ↔ 𝑠 < (abs‘𝐷)))
99		oveq2 5952	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝑞 · 𝐷) + 𝑟) = ((𝑞 · 𝐷) + 𝑠))
100	99	eqeq2d 2217	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑁 = ((𝑞 · 𝐷) + 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑠)))
101	97, 98, 100	3anbi123d 1325	. . . . 5 ⊢ (𝑟 = 𝑠 → ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))))
102	101	rexbidv 2507	. . . 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))))
103	102	rmo4 2966	. . 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))
104	96, 103	sylibr 134	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃*𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
105		reu5 2723	. 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))))
106	4, 104, 105	sylanbrc 417	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ w3o 980 ∧ w3a 981 = wceq 1373 ∈ wcel 2176 ≠ wne 2376 ∀wral 2484 ∃wrex 2485 ∃!wreu 2486 ∃*wrmo 2487 class class class wbr 4044 ‘cfv 5271 (class class class)co 5944 0cc0 7925 + caddc 7928 · cmul 7930 < clt 8107 ≤ cle 8108 -cneg 8244 ℕcn 9036 ℤcz 9372 abscabs 11308

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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043 ax-arch 8044

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-rp 9776 df-fl 10413 df-mod 10468 df-seqfrec 10593 df-exp 10684 df-cj 11153 df-re 11154 df-im 11155 df-rsqrt 11309 df-abs 11310

This theorem is referenced by: divalg 12235

W3C validator