Theorem nndiffz1 32690

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 12637	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
2		nn0z 12628	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		elfz1 13536	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
4	1, 2, 3	sylancr 585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
5		3anass 1092	. . . . . . . . . . 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 538	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
8	7	baibd 538	. . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℤ)
11	10	zred 12711	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℝ)
12		simpr 483	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
13	12	zred 12711	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
14	11, 13	ltnled 11401	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
15		zltp1le 12657	. . . . . . . . . 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 578	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
18	17	adantr 479	. . . . . . 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 577	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
21		1red 11255	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ∈ ℝ)
22		simpll 765	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℕ0)
23	22	nn0red 12578	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℝ)
24	23, 21	readdcld 11283	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ∈ ℝ)
25		simplr 767	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℤ)
26	25	zred 12711	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℝ)
27		0p1e1 12379	. . . . . . . . 9 ⊢ (0 + 1) = 1
28		0red 11257	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ∈ ℝ)
29	22	nn0ge0d 12580	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ≤ 𝑁)
30	28, 23, 21, 29	leadd1dd 11868	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (0 + 1) ≤ (𝑁 + 1))
31	27, 30	eqbrtrrid 5181	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ (𝑁 + 1))
32		simpr 483	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ≤ 𝑗)
33	21, 24, 26, 31, 32	letrd 11411	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ 𝑗)
34	33	ex 411	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑗 → 1 ≤ 𝑗))
35	34	pm4.71rd 561	. . . . 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 577	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
38		eldif 3958	. . . . 5 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)))
39		elnnz1 12633	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
40	39	anbi1i 622	. . . . 5 ⊢ ((𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗) ∧ ¬ 𝑗 ∈ (1...𝑁)))
41		anass 467	. . . . 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 12557	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
45	44	nn0zd 12629	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ)
46		eluz1 12871	. . . 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 2724	1 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∖ cdif 3945   class class class wbr 5145  ‘cfv 6545  (class class class)co 7415  0cc0 11148  1c1 11149   + caddc 11151   < clt 11288   ≤ cle 11289  ℕcn 12257  ℕ₀cn0 12517  ℤcz 12603  ℤ_≥cuz 12867  ...cfz 13531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3968  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-en 8966  df-dom 8967  df-sdom 8968  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12258  df-n0 12518  df-z 12604  df-uz 12868  df-fz 13532

This theorem is referenced by:  eulerpartlems  34206  eulerpartlemsv3  34207  eulerpartlemgc  34208