Theorem fzsplit1nn0 42786

Description: Split a finite 1-based set of integers in the middle, allowing either end to be empty ((1...0)). (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
fzsplit1nn0	⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

Proof of Theorem fzsplit1nn0

Step	Hyp	Ref	Expression
1		elnn0 12380	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		1zzd 12500	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ∈ ℤ)
3		nn0z 12490	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ)
4	3	ad2antrl 728	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐵 ∈ ℤ)
5		nnz 12486	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℤ)
7		nnge1 12150	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
8	7	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ≤ 𝐴)
9		simprr 772	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
10	2, 4, 6, 8, 9	elfzd 13412	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ (1...𝐵))
11		fzsplit 13447	. . . . . 6 ⊢ (𝐴 ∈ (1...𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
12	10, 11	syl 17	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
13		uncom 4108	. . . . . 6 ⊢ ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
14		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
15	14	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = (0 + 1))
16		0p1e1 12239	. . . . . . . . . 10 ⊢ (0 + 1) = 1
17	15, 16	eqtrdi 2782	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = 1)
18	17	oveq1d 7361	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
19		oveq2 7354	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0))
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = (1...0))
21		fz10 13442	. . . . . . . . 9 ⊢ (1...0) = ∅
22	20, 21	eqtrdi 2782	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = ∅)
23	18, 22	uneq12d 4119	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
24		un0 4344	. . . . . . 7 ⊢ ((1...𝐵) ∪ ∅) = (1...𝐵)
25	23, 24	eqtrdi 2782	. . . . . 6 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
26	13, 25	eqtr2id 2779	. . . . 5 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
27	12, 26	jaoian 958	. . . 4 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
28	27	ex 412	. . 3 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → ((𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
29	1, 28	sylbi 217	. 2 ⊢ (𝐴 ∈ ℕ0 → ((𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
30	29	3impib 1116	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∪ cun 3900  ∅c0 4283   class class class wbr 5091  (class class class)co 7346  0cc0 11003  1c1 11004   + caddc 11006   ≤ cle 11144  ℕcn 12122  ℕ₀cn0 12378  ℤcz 12465  ...cfz 13404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-n0 12379  df-z 12466  df-uz 12730  df-fz 13405

This theorem is referenced by:  eldioph2lem1  42792