Theorem fzsplit1nn0 40231

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 12075	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		1zzd 12191	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ∈ ℤ)
3		nn0z 12183	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ)
4	3	ad2antrl 728	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐵 ∈ ℤ)
5		nnz 12182	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
6	5	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℤ)
7		nnge1 11841	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
8	7	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ≤ 𝐴)
9		simprr 773	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
10	2, 4, 6, 8, 9	elfzd 13086	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ (1...𝐵))
11		fzsplit 13121	. . . . . 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 4057	. . . . . 6 ⊢ ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
14		oveq1 7209	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
15	14	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = (0 + 1))
16		0p1e1 11935	. . . . . . . . . 10 ⊢ (0 + 1) = 1
17	15, 16	eqtrdi 2790	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = 1)
18	17	oveq1d 7217	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
19		oveq2 7210	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0))
20	19	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = (1...0))
21		fz10 13116	. . . . . . . . 9 ⊢ (1...0) = ∅
22	20, 21	eqtrdi 2790	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = ∅)
23	18, 22	uneq12d 4068	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
24		un0 4295	. . . . . . 7 ⊢ ((1...𝐵) ∪ ∅) = (1...𝐵)
25	23, 24	eqtrdi 2790	. . . . . 6 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
26	13, 25	eqtr2id 2787	. . . . 5 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
27	12, 26	jaoian 957	. . . 4 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
28	27	ex 416	. . 3 ⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → ((𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
29	1, 28	sylbi 220	. 2 ⊢ (𝐴 ∈ ℕ0 → ((𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
30	29	3impib 1118	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110   ∪ cun 3855  ∅c0 4227   class class class wbr 5043  (class class class)co 7202  0cc0 10712  1c1 10713   + caddc 10715   ≤ cle 10851  ℕcn 11813  ℕ₀cn0 12073  ℤcz 12159  ...cfz 13078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512  ax-cnex 10768  ax-resscn 10769  ax-1cn 10770  ax-icn 10771  ax-addcl 10772  ax-addrcl 10773  ax-mulcl 10774  ax-mulrcl 10775  ax-mulcom 10776  ax-addass 10777  ax-mulass 10778  ax-distr 10779  ax-i2m1 10780  ax-1ne0 10781  ax-1rid 10782  ax-rnegex 10783  ax-rrecex 10784  ax-cnre 10785  ax-pre-lttri 10786  ax-pre-lttrn 10787  ax-pre-ltadd 10788  ax-pre-mulgt0 10789

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-riota 7159  df-ov 7205  df-oprab 7206  df-mpo 7207  df-om 7634  df-1st 7750  df-2nd 7751  df-wrecs 8036  df-recs 8097  df-rdg 8135  df-er 8380  df-en 8616  df-dom 8617  df-sdom 8618  df-pnf 10852  df-mnf 10853  df-xr 10854  df-ltxr 10855  df-le 10856  df-sub 11047  df-neg 11048  df-nn 11814  df-n0 12074  df-z 12160  df-uz 12422  df-fz 13079

This theorem is referenced by:  eldioph2lem1  40237