Theorem nndiffz1 32763

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 12622	. . . . . . . . . . . 12 ⊢ 1 ∈ ℤ
2		nn0z 12613	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
3		elfz1 13529	. . . . . . . . . . . 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 12697	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑁 ∈ ℝ)
12		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℤ)
13	12	zred 12697	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → 𝑗 ∈ ℝ)
14	11, 13	ltnled 11382	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑁 < 𝑗 ↔ ¬ 𝑗 ≤ 𝑁))
15		zltp1le 12642	. . . . . . . . . 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 11236	. . . . . . . 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 12563	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑁 ∈ ℝ)
24	23, 21	readdcld 11264	. . . . . . . 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 12697	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 𝑗 ∈ ℝ)
27		0p1e1 12362	. . . . . . . . 9 ⊢ (0 + 1) = 1
28		0red 11238	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ∈ ℝ)
29	22	nn0ge0d 12565	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → 0 ≤ 𝑁)
30	28, 23, 21, 29	leadd1dd 11851	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑗 ∈ ℤ) ∧ (𝑁 + 1) ≤ 𝑗) → (0 + 1) ≤ (𝑁 + 1))
31	27, 30	eqbrtrrid 5155	. . . . . . . 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 11392	. . . . . . 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 3936	. . . . 5 ⊢ (𝑗 ∈ (ℕ ∖ (1...𝑁)) ↔ (𝑗 ∈ ℕ ∧ ¬ 𝑗 ∈ (1...𝑁)))
39		elnnz1 12618	. . . . . 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 12541	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
45	44	nn0zd 12614	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℤ)
46		eluz1 12856	. . . 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 2733	1 ⊢ (𝑁 ∈ ℕ0 → (ℕ ∖ (1...𝑁)) = (ℤ≥‘(𝑁 + 1)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ∖ cdif 3923   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  0cc0 11129  1c1 11130   + caddc 11132   < clt 11269   ≤ cle 11270  ℕcn 12240  ℕ₀cn0 12501  ℤcz 12588  ℤ_≥cuz 12852  ...cfz 13524

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-fz 13525

This theorem is referenced by:  eulerpartlems  34392  eulerpartlemsv3  34393  eulerpartlemgc  34394