Theorem fzsplit1nn0 43468

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 12507	. . 3 ⊢ (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2		1zzd 12626	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ∈ ℤ)
3		nn0z 12616	. . . . . . . 8 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℤ)
4	3	ad2antrl 740	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐵 ∈ ℤ)
5		nnz 12613	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
6	5	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℤ)
7		nnge1 12265	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 1 ≤ 𝐴)
8	7	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 1 ≤ 𝐴)
9		simprr 784	. . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ≤ 𝐵)
10	2, 4, 6, 8, 9	elfzd 13544	. . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ (1...𝐵))
11		fzsplit 13580	. . . . . 6 ⊢ (𝐴 ∈ (1...𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
12	10, 11	syl 18	. . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
13		uncom 4113	. . . . . 6 ⊢ ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
14		oveq1 7419	. . . . . . . . . . 11 ⊢ (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
15	14	adantr 485	. . . . . . . . . 10 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = (0 + 1))
16		0p1e1 12362	. . . . . . . . . 10 ⊢ (0 + 1) = 1
17	15, 16	eqtrdi 2814	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (𝐴 + 1) = 1)
18	17	oveq1d 7427	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
19		oveq2 7420	. . . . . . . . . 10 ⊢ (𝐴 = 0 → (1...𝐴) = (1...0))
20	19	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = (1...0))
21		fz10 13574	. . . . . . . . 9 ⊢ (1...0) = ∅
22	20, 21	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐴) = ∅)
23	18, 22	uneq12d 4124	. . . . . . 7 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
24		un0 4352	. . . . . . 7 ⊢ ((1...𝐵) ∪ ∅) = (1...𝐵)
25	23, 24	eqtrdi 2814	. . . . . 6 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
26	13, 25	eqtr2id 2811	. . . . 5 ⊢ ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
27	12, 26	jaoian 971	. . . 4 ⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
28	27	ex 417	. . 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 1134	1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0 ∧ 𝐴 ≤ 𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ∪ cun 3904  ∅c0 4287   class class class wbr 5110  (class class class)co 7412  0cc0 11101  1c1 11102   + caddc 11104   ≤ cle 11245  ℕcn 12234  ℕ₀cn0 12505  ℤcz 12592  ...cfz 13536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537

This theorem is referenced by:  eldioph2lem1  43474