Theorem nndiffz1 29548

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 11407	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
2		nn0z 11400	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		elfz1 12331	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
4	1, 2, 3	sylancr 695	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
5		3anass 1042	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
6	4, 5	syl6bb 276	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (1...𝑁) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁))))
7	6	baibd 948	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (𝑗 ∈ (1...𝑁) ↔ (1 ≤ 𝑗 ∧ 𝑗 ≤ 𝑁)))
8	7	baibd 948	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (𝑗 ∈ (1...𝑁) ↔ 𝑗 ≤ 𝑁))
9	8	notbid 308	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ∈ (1...𝑁) ↔ ¬ 𝑗 ≤ 𝑁))
10		simpl 473	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℤ)
11	10	zred 11482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℝ)
12		simpr 477	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
13	12	zred 11482	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
14	11, 13	ltnled 10184	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
15		zltp1le 11427	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ (𝑁 + 1) ≤ 𝑗))
16	14, 15	bitr3d 270	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
17	2, 16	sylan 488	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
18	17	adantr 481	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ≤ 𝑁 ↔ (𝑁 + 1) ≤ 𝑗))
19	9, 18	bitrd 268	. . . . . 6 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ 1 ≤ 𝑗) → (¬ 𝑗 ∈ (1...𝑁) ↔ (𝑁 + 1) ≤ 𝑗))
20	19	pm5.32da 673	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
21		1red 10055	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ∈ ℝ)
22		simpll 790	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℕ0)
23	22	nn0red 11352	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℝ)
24	23, 21	readdcld 10069	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ∈ ℝ)
25		simplr 792	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℤ)
26	25	zred 11482	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℝ)
27		0p1e1 11132	. . . . . . . . 9 ⊢ (0 + 1) = 1
28		0red 10041	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ∈ ℝ)
29	22	nn0ge0d 11354	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ≤ 𝑁)
30	28, 23, 21, 29	leadd1dd 10641	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (0 + 1) ≤ (𝑁 + 1))
31	27, 30	syl5eqbrr 4689	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ (𝑁 + 1))
32		simpr 477	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (𝑁 + 1) ≤ 𝑗)
33	21, 24, 26, 31, 32	letrd 10194	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 1 ≤ 𝑗)
34	33	ex 450	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑗 → 1 ≤ 𝑗))
35	34	pm4.71rd 667	. . . . 5 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((𝑁 + 1) ≤ 𝑗 ↔ (1 ≤ 𝑗 ∧ (𝑁 + 1) ≤ 𝑗)))
36	20, 35	bitr4d 271	. . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) → ((1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (𝑁 + 1) ≤ 𝑗))
37	36	pm5.32da 673	. . 3 ⊢ (𝑁 ∈ ℕ0 → ((𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))) ↔ (𝑗 ∈ ℤ ∧ (𝑁 + 1) ≤ 𝑗)))
38		eldif 3584	. . . . 5 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)))
39		elnnz1 11403	. . . . . 6 ⊢ (𝑗 ∈ ℕ ↔ (𝑗 ∈ ℤ ∧ 1 ≤ 𝑗))
40	39	anbi1i 731	. . . . 5 ⊢ ((𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ ((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗) ∧ ¬ 𝑗 ∈ (1...𝑁)))
41		anass 681	. . . . 5 ⊢ (((𝑗 ∈ ℤ ∧ 1 ≤ 𝑗) ∧ ¬ 𝑗 ∈ (1...𝑁)) ↔ (𝑗 ∈ ℤ ∧ (1 ≤ 𝑗 ∧ ¬ 𝑗 ∈ (1...𝑁))))
42	38, 40, 41	3bitri 286	. . . 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 11333	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
45	44	nn0zd 11480	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ)
46		eluz1 11691	. . . 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 300	. 2 ⊢ (𝑁 ∈ ℕ0 → (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ 𝑗 ∈ (ℤ≥‘(𝑁 + 1))))
49	48	eqrdv 2620	1 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∖ cdif 3571   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  eulerpartlems  30422  eulerpartlemsv3  30423  eulerpartlemgc  30424