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Theorem divalgb 12566

Description: Express the division algorithm as stated in divalg 12565 in terms of ∥. (Contributed by Paul Chapman, 31-Mar-2011.)

**Assertion**
Ref	Expression
divalgb	⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ₀ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))

Distinct variable groups: 𝐷,𝑞,𝑟 𝑁,𝑞,𝑟

**Proof of Theorem divalgb**
Step	Hyp	Ref	Expression
1		df-3an 1007	. . . . . . . . 9 ⊢ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
2	1	rexbii 2540	. . . . . . . 8 ⊢ (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃𝑞 ∈ ℤ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
3		r19.42v 2691	. . . . . . . 8 ⊢ (∃𝑞 ∈ ℤ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
4	2, 3	bitri 184	. . . . . . 7 ⊢ (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
5		zsubcl 9581	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (𝑁 − 𝑟) ∈ ℤ)
6		divides 12430	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 − 𝑟) ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟)))
7	5, 6	sylan2 286	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ ℤ ∧ (𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ)) → (𝐷 ∥ (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟)))
8	7	3impb 1226	. . . . . . . . . 10 ⊢ ((𝐷 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟)))
9	8	3com12 1234	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟)))
10		zcn 9545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
11		zcn 9545	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℤ → 𝑟 ∈ ℂ)
12		zmulcl 9594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝑞 · 𝐷) ∈ ℤ)
13	12	zcnd 9664	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝑞 · 𝐷) ∈ ℂ)
14		subadd 8441	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℂ ∧ 𝑟 ∈ ℂ ∧ (𝑞 · 𝐷) ∈ ℂ) → ((𝑁 − 𝑟) = (𝑞 · 𝐷) ↔ (𝑟 + (𝑞 · 𝐷)) = 𝑁))
15	10, 11, 13, 14	syl3an 1316	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝑁 − 𝑟) = (𝑞 · 𝐷) ↔ (𝑟 + (𝑞 · 𝐷)) = 𝑁))
16		addcom 8375	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑟 ∈ ℂ ∧ (𝑞 · 𝐷) ∈ ℂ) → (𝑟 + (𝑞 · 𝐷)) = ((𝑞 · 𝐷) + 𝑟))
17	11, 13, 16	syl2an 289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝑟 + (𝑞 · 𝐷)) = ((𝑞 · 𝐷) + 𝑟))
18	17	3adant1 1042	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → (𝑟 + (𝑞 · 𝐷)) = ((𝑞 · 𝐷) + 𝑟))
19	18	eqeq1d 2240	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝑟 + (𝑞 · 𝐷)) = 𝑁 ↔ ((𝑞 · 𝐷) + 𝑟) = 𝑁))
20	15, 19	bitrd 188	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝑁 − 𝑟) = (𝑞 · 𝐷) ↔ ((𝑞 · 𝐷) + 𝑟) = 𝑁))
21		eqcom 2233	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 − 𝑟) = (𝑞 · 𝐷) ↔ (𝑞 · 𝐷) = (𝑁 − 𝑟))
22		eqcom 2233	. . . . . . . . . . . . . . . 16 ⊢ (((𝑞 · 𝐷) + 𝑟) = 𝑁 ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟))
23	20, 21, 22	3bitr3g 222	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ (𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ((𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
24	23	3expia 1232	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ) → ((𝑞 ∈ ℤ ∧ 𝐷 ∈ ℤ) → ((𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟))))
25	24	expcomd 1487	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (𝐷 ∈ ℤ → (𝑞 ∈ ℤ → ((𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))))
26	25	3impia 1227	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (𝑞 ∈ ℤ → ((𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟))))
27	26	imp 124	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ 𝑞 ∈ ℤ) → ((𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
28	27	rexbidva 2530	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑟 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
29	28	3com23 1236	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (∃𝑞 ∈ ℤ (𝑞 · 𝐷) = (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
30	9, 29	bitrd 188	. . . . . . . 8 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (𝐷 ∥ (𝑁 − 𝑟) ↔ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟)))
31	30	anbi2d 464	. . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝐷 ∥ (𝑁 − 𝑟)) ↔ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ ∃𝑞 ∈ ℤ 𝑁 = ((𝑞 · 𝐷) + 𝑟))))
32	4, 31	bitr4id 199	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝐷 ∥ (𝑁 − 𝑟))))
33		anass 401	. . . . . 6 ⊢ (((0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷)) ∧ 𝐷 ∥ (𝑁 − 𝑟)) ↔ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))
34	32, 33	bitrdi 196	. . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝑟 ∈ ℤ) → (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
35	34	3expa 1230	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ) ∧ 𝑟 ∈ ℤ) → (∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
36	35	reubidva 2718	. . 3 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℤ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
37		elnn0z 9553	. . . . . . 7 ⊢ (𝑟 ∈ ℕ₀ ↔ (𝑟 ∈ ℤ ∧ 0 ≤ 𝑟))
38	37	anbi1i 458	. . . . . 6 ⊢ ((𝑟 ∈ ℕ₀ ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ ((𝑟 ∈ ℤ ∧ 0 ≤ 𝑟) ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))
39		anass 401	. . . . . 6 ⊢ (((𝑟 ∈ ℤ ∧ 0 ≤ 𝑟) ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ (𝑟 ∈ ℤ ∧ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
40	38, 39	bitri 184	. . . . 5 ⊢ ((𝑟 ∈ ℕ₀ ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ (𝑟 ∈ ℤ ∧ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
41	40	eubii 2088	. . . 4 ⊢ (∃!𝑟(𝑟 ∈ ℕ₀ ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ ∃!𝑟(𝑟 ∈ ℤ ∧ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
42		df-reu 2518	. . . 4 ⊢ (∃!𝑟 ∈ ℕ₀ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)) ↔ ∃!𝑟(𝑟 ∈ ℕ₀ ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))
43		df-reu 2518	. . . 4 ⊢ (∃!𝑟 ∈ ℤ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ ∃!𝑟(𝑟 ∈ ℤ ∧ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))))
44	41, 42, 43	3bitr4ri 213	. . 3 ⊢ (∃!𝑟 ∈ ℤ (0 ≤ 𝑟 ∧ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))) ↔ ∃!𝑟 ∈ ℕ₀ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟)))
45	36, 44	bitrdi 196	. 2 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ₀ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))
46	45	3adant3 1044	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝐷 ∈ ℤ ∧ 𝐷 ≠ 0) → (∃!𝑟 ∈ ℤ ∃𝑞 ∈ ℤ (0 ≤ 𝑟 ∧ 𝑟 < (abs‘𝐷) ∧ 𝑁 = ((𝑞 · 𝐷) + 𝑟)) ↔ ∃!𝑟 ∈ ℕ₀ (𝑟 < (abs‘𝐷) ∧ 𝐷 ∥ (𝑁 − 𝑟))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∃!weu 2079 ∈ wcel 2202 ≠ wne 2403 ∃wrex 2512 ∃!wreu 2513 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8090 0cc0 8092 + caddc 8095 · cmul 8097 < clt 8273 ≤ cle 8274 − cmin 8409 ℕ₀cn0 9461 ℤcz 9540 abscabs 11637 ∥ cdvds 12428

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-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-cnex 8183 ax-resscn 8184 ax-1cn 8185 ax-1re 8186 ax-icn 8187 ax-addcl 8188 ax-addrcl 8189 ax-mulcl 8190 ax-mulrcl 8191 ax-addcom 8192 ax-mulcom 8193 ax-addass 8194 ax-mulass 8195 ax-distr 8196 ax-i2m1 8197 ax-0lt1 8198 ax-1rid 8199 ax-0id 8200 ax-rnegex 8201 ax-cnre 8203 ax-pre-ltirr 8204 ax-pre-ltwlin 8205 ax-pre-lttrn 8206 ax-pre-ltadd 8208

This theorem depends on definitions: df-bi 117 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-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-pnf 8275 df-mnf 8276 df-xr 8277 df-ltxr 8278 df-le 8279 df-sub 8411 df-neg 8412 df-inn 9203 df-n0 9462 df-z 9541 df-dvds 12429

This theorem is referenced by: divalg2 12567

W3C validator