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Theorem fzsplit1nn0 36335
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 ∈ ℕ0𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))

Proof of Theorem fzsplit1nn0
StepHypRef Expression
1 elnn0 11171 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 nnge1 10923 . . . . . . . 8 (𝐴 ∈ ℕ → 1 ≤ 𝐴)
32adantr 480 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ≤ 𝐴)
4 simprr 792 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴𝐵)
5 nnz 11276 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
65adantr 480 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ ℤ)
7 1zzd 11285 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ∈ ℤ)
8 nn0z 11277 . . . . . . . . 9 (𝐵 ∈ ℕ0𝐵 ∈ ℤ)
98ad2antrl 760 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐵 ∈ ℤ)
10 elfz 12203 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
116, 7, 9, 10syl3anc 1318 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
123, 4, 11mpbir2and 959 . . . . . 6 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ (1...𝐵))
13 fzsplit 12238 . . . . . 6 (𝐴 ∈ (1...𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
1412, 13syl 17 . . . . 5 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
15 uncom 3719 . . . . . 6 ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
16 oveq1 6556 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
1716adantr 480 . . . . . . . . . 10 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = (0 + 1))
18 0p1e1 11009 . . . . . . . . . 10 (0 + 1) = 1
1917, 18syl6eq 2660 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = 1)
2019oveq1d 6564 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
21 oveq2 6557 . . . . . . . . . 10 (𝐴 = 0 → (1...𝐴) = (1...0))
2221adantr 480 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = (1...0))
23 fz10 12233 . . . . . . . . 9 (1...0) = ∅
2422, 23syl6eq 2660 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = ∅)
2520, 24uneq12d 3730 . . . . . . 7 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
26 un0 3919 . . . . . . 7 ((1...𝐵) ∪ ∅) = (1...𝐵)
2725, 26syl6eq 2660 . . . . . 6 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
2815, 27syl5req 2657 . . . . 5 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
2914, 28jaoian 820 . . . 4 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
3029ex 449 . . 3 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
311, 30sylbi 206 . 2 (𝐴 ∈ ℕ0 → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
32313impib 1254 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  cun 3538  c0 3874   class class class wbr 4583  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  cle 9954  cn 10897  0cn0 11169  cz 11254  ...cfz 12197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198
This theorem is referenced by:  eldioph2lem1  36341
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