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Theorem divalglemeuneg 11882

Description: Lemma for divalg 11883. 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 994	. . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 < 0)
2	1	lt0ne0d 8432	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ≠ 0)
3		divalglemex 11881	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
4	2, 3	syld3an3 1278	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
5		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑞((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ))
6		nfre1 2513	. . . . . . 7 ⊢ Ⅎ𝑞∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))
7		nfv 1521	. . . . . . 7 ⊢ Ⅎ𝑞 𝑟 = 𝑠
8	6, 7	nfim 1565	. . . . . 6 ⊢ Ⅎ𝑞(∃𝑞 ∈ ℤ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) → 𝑟 = 𝑠)
9		oveq1 5860	. . . . . . . . . . . 12 ⊢ (𝑞 = 𝑡 → (𝑞 · 𝐷) = (𝑡 · 𝐷))
10	9	oveq1d 5868	. . . . . . . . . . 11 ⊢ (𝑞 = 𝑡 → ((𝑞 · 𝐷) + 𝑠) = ((𝑡 · 𝐷) + 𝑠))
11	10	eqeq2d 2182	. . . . . . . . . 10 ⊢ (𝑞 = 𝑡 → (𝑁 = ((𝑞 · 𝐷) + 𝑠) ↔ 𝑁 = ((𝑡 · 𝐷) + 𝑠)))
12	11	3anbi3d 1313	. . . . . . . . 9 ⊢ (𝑞 = 𝑡 → ((0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑡 · 𝐷) + 𝑠))))
13	12	cbvrexv 2697	. . . . . . . 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 109	. . . . . . . . . . . 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 993	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ∈ ℤ)
16	15	znegcld 9336	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → -𝐷 ∈ ℤ)
17	15	zred 9334	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 𝐷 ∈ ℝ)
18	17	lt0neg1d 8434	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → (𝐷 < 0 ↔ 0 < -𝐷))
19	1, 18	mpbid 146	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → 0 < -𝐷)
20		elnnz 9222	. . . . . . . . . . . . . . . . 17 ⊢ (-𝐷 ∈ ℕ ↔ (-𝐷 ∈ ℤ ∧ 0 < -𝐷))
21	16, 19, 20	sylanbrc 415	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → -𝐷 ∈ ℕ)
22	21	ad5antr 493	. . . . . . . . . . . . . . 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 531	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑠 ∈ ℤ)
24	23	ad3antrrr 489	. . . . . . . . . . . . . . 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 530	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑟 ∈ ℤ)
26	25	ad3antrrr 489	. . . . . . . . . . . . . . 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 525	. . . . . . . . . . . . . . . 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 9336	. . . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝑞 ∈ ℤ)
30	29	ad3antrrr 489	. . . . . . . . . . . . . . . 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 9336	. . . . . . . . . . . . . . 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 998	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 999	. . . . . . . . . . . . . . . 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 1032	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝐷 ∈ ℤ)
36	35	ad3antrrr 489	. . . . . . . . . . . . . . . . . 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 9334	. . . . . . . . . . . . . . . . 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 7921	. . . . . . . . . . . . . . . . . 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 1033	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) ∧ (𝑟 ∈ ℤ ∧ 𝑠 ∈ ℤ)) ∧ 𝑞 ∈ ℤ) → 𝐷 < 0)
40	39	ad3antrrr 489	. . . . . . . . . . . . . . . . . 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 8038	. . . . . . . . . . . . . . . . 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 11124	. . . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . 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 1000	. . . . . . . . . . . . . . . . 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 9335	. . . . . . . . . . . . . . . . . . 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 9335	. . . . . . . . . . . . . . . . . . 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 8332	. . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . 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 2206	. . . . . . . . . . . . . . . 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 1000	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . 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 9335	. . . . . . . . . . . . . . . . . . 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 8332	. . . . . . . . . . . . . . . . . 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 5868	. . . . . . . . . . . . . . . . 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 2206	. . . . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . . . 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 11879	. . . . . . . . . . . . . 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 9334	. . . . . . . . . . . . . . 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 9334	. . . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . 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 668	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 615	. . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 274	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 2205	. . . . . . . . . . . 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 11878	. . . . . . . . . . 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 109	. . . . . . . . . . . 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 998	. . . . . . . . . . . . . . 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 999	. . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . 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 2205	. . . . . . . . . . . . . . 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 11879	. . . . . . . . . . . . . 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 8442	. . . . . . . . . . . . . 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 668	. . . . . . . . . . . . 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 274	. . . . . . . . . . . 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 615	. . . . . . . . . . 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 525	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 9255	. . . . . . . . . . . 12 ⊢ ((𝑞 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑞 < 𝑡 ∨ 𝑞 = 𝑡 ∨ 𝑡 < 𝑞))
88	86, 27, 87	syl2anc 409	. . . . . . . . . . 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 1302	. . . . . . . . . 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 114	. . . . . . . . 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 2587	. . . . . . . 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	syl5bi 151	. . . . . . 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 362	. . . . . 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 2584	. . . . 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 252	. . . 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 2552	. . 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 3993	. . . . . 6 ⊢ (𝑟 = 𝑠 → (0 ≤ 𝑟 ↔ 0 ≤ 𝑠))
98		breq1 3992	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 < (abs‘𝐷) ↔ 𝑠 < (abs‘𝐷)))
99		oveq2 5861	. . . . . . 7 ⊢ (𝑟 = 𝑠 → ((𝑞 · 𝐷) + 𝑟) = ((𝑞 · 𝐷) + 𝑠))
100	99	eqeq2d 2182	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑁 = ((𝑞 · 𝐷) + 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑠)))
101	97, 98, 100	3anbi123d 1307	. . . . 5 ⊢ (𝑟 = 𝑠 → ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (0 ≤ 𝑠 ∧ 𝑠 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑠))))
102	101	rexbidv 2471	. . . 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 2923	. . 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 133	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 < 0) → ∃*𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
105		reu5 2682	. 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 415	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 103 ∨ w3o 972 ∧ w3a 973 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 ∃wrex 2449 ∃!wreu 2450 ∃*wrmo 2451 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 0cc0 7774 + caddc 7777 · cmul 7779 < clt 7954 ≤ cle 7955 -cneg 8091 ℕcn 8878 ℤcz 9212 abscabs 10961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963

This theorem is referenced by: divalg 11883

W3C validator