Theorem nndiffz1 31009

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 12280	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
2		nn0z 12273	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		elfz1 13173	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
4	1, 2, 3	sylancr 586	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
5		3anass 1093	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
6	4, 5	bitrdi 286	. . . . . . . . . 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 317	. . . . . . 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 12355	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℝ)
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
13	12	zred 12355	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
14	11, 13	ltnled 11052	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
15		zltp1le 12300	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ (𝑁 + 1) ≤ 𝑗))
16	14, 15	bitr3d 280	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
17	2, 16	sylan 579	. . . . . . . 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 278	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ∈ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑗))
20	19	pm5.32da 578	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
21		1red 10907	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ∈ ℝ)
22		simpll 763	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℕ0)
23	22	nn0red 12224	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℝ)
24	23, 21	readdcld 10935	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ∈ ℝ)
25		simplr 765	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℤ)
26	25	zred 12355	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℝ)
27		0p1e1 12025	. . . . . . . . 9 ⊢ (0 + 1) = 1
28		0red 10909	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ∈ ℝ)
29	22	nn0ge0d 12226	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ≤ 𝑁)
30	28, 23, 21, 29	leadd1dd 11519	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (0 + 1) ≤ (𝑁 + 1))
31	27, 30	eqbrtrrid 5106	. . . . . . . 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 11062	. . . . . . 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 281	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑗))
37	36	pm5.32da 578	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
38		eldif 3893	. . . . 5 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)))
39		elnnz1 12276	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
40	39	anbi1i 623	. . . . 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 296	. . . 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 12203	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
45	44	nn0zd 12353	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ)
46		eluz1 12515	. . . 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 310	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))))
49	48	eqrdv 2736	1 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ∖ cdif 3880   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803   + caddc 10805   < clt 10940   ≤ cle 10941  ℕcn 11903  ℕ₀cn0 12163  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169

This theorem is referenced by:  eulerpartlems  32227  eulerpartlemsv3  32228  eulerpartlemgc  32229