Theorem nndiffz1 32738

Description: Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)

Assertion

Ref	Expression
nndiffz1	⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Proof of Theorem nndiffz1

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1z 12505	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
2		nn0z 12496	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		elfz1 13415	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
4	1, 2, 3	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
5		3anass 1094	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
6	4, 5	bitrdi 287	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
7	6	baibd 539	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
8	7	baibd 539	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (𝑗 ∈ (1...𝑁) ↔ 𝑗 ≤ 𝑁))
9	8	notbid 318	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ∈ (1...𝑁) ↔ ¬ 𝑗 ≤ 𝑁))
10		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℤ)
11	10	zred 12580	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℝ)
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
13	12	zred 12580	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
14	11, 13	ltnled 11263	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
15		zltp1le 12525	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ (𝑁 + 1) ≤ 𝑗))
16	14, 15	bitr3d 281	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
17	2, 16	sylan 580	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
18	17	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
19	9, 18	bitrd 279	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ∈ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑗))
20	19	pm5.32da 579	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
21		1red 11116	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ∈ ℝ)
22		simpll 766	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℕ0)
23	22	nn0red 12446	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℝ)
24	23, 21	readdcld 11144	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ∈ ℝ)
25		simplr 768	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℤ)
26	25	zred 12580	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℝ)
27		0p1e1 12245	. . . . . . . . 9 ⊢ (0 + 1) = 1
28		0red 11118	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ∈ ℝ)
29	22	nn0ge0d 12448	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ≤ 𝑁)
30	28, 23, 21, 29	leadd1dd 11734	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (0 + 1) ≤ (𝑁 + 1))
31	27, 30	eqbrtrrid 5128	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ (𝑁 + 1))
32		simpr 484	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ≤ 𝑗)
33	21, 24, 26, 31, 32	letrd 11273	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ 𝑗)
34	33	ex 412	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑗 → 1 ≤ 𝑗))
35	34	pm4.71rd 562	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑗 ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
36	20, 35	bitr4d 282	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑗))
37	36	pm5.32da 579	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
38		eldif 3913	. . . . 5 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)))
39		elnnz1 12501	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
40	39	anbi1i 624	. . . . 5 ⊢ ((𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗) ∧ ¬ 𝑗 ∈ (1...𝑁)))
41		anass 468	. . . . 5 ⊢ (((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗) ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))))
42	38, 40, 41	3bitri 297	. . . 4 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))))
43	42	a1i 11	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)))))
44		peano2nn0 12424	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
45	44	nn0zd 12497	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ)
46		eluz1 12739	. . . 4 ⊢ ((𝑁 + 1) ∈ ℤ → (𝑗 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
47	45, 46	syl 17	. . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (ℤ≥‘(𝑁 + 1)) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
48	37, 43, 47	3bitr4d 311	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))))
49	48	eqrdv 2727	1 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  0cc0 11009  1c1 11010   + caddc 11012   < clt 11149   ≤ cle 11150  ℕcn 12128  ℕ₀cn0 12384  ℤcz 12471  ℤ_≥cuz 12735  ...cfz 13410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411

This theorem is referenced by:  eulerpartlems  34344  eulerpartlemsv3  34345  eulerpartlemgc  34346