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Theorem fzsplit1nn0 36131
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 11141 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 nnge1 10893 . . . . . . . 8 (𝐴 ∈ ℕ → 1 ≤ 𝐴)
32adantr 479 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ≤ 𝐴)
4 simprr 791 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴𝐵)
5 nnz 11232 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
65adantr 479 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ ℤ)
7 1zzd 11241 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ∈ ℤ)
8 nn0z 11233 . . . . . . . . 9 (𝐵 ∈ ℕ0𝐵 ∈ ℤ)
98ad2antrl 759 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐵 ∈ ℤ)
10 elfz 12158 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
116, 7, 9, 10syl3anc 1317 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
123, 4, 11mpbir2and 958 . . . . . 6 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ (1...𝐵))
13 fzsplit 12193 . . . . . 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 3718 . . . . . 6 ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
16 oveq1 6534 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
1716adantr 479 . . . . . . . . . 10 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = (0 + 1))
18 0p1e1 10979 . . . . . . . . . 10 (0 + 1) = 1
1917, 18syl6eq 2659 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = 1)
2019oveq1d 6542 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
21 oveq2 6535 . . . . . . . . . 10 (𝐴 = 0 → (1...𝐴) = (1...0))
2221adantr 479 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = (1...0))
23 fz10 12188 . . . . . . . . 9 (1...0) = ∅
2422, 23syl6eq 2659 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = ∅)
2520, 24uneq12d 3729 . . . . . . 7 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
26 un0 3918 . . . . . . 7 ((1...𝐵) ∪ ∅) = (1...𝐵)
2725, 26syl6eq 2659 . . . . . 6 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
2815, 27syl5req 2656 . . . . 5 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
2914, 28jaoian 819 . . . 4 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
3029ex 448 . . 3 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
311, 30sylbi 205 . 2 (𝐴 ∈ ℕ0 → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
32313impib 1253 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382  w3a 1030   = wceq 1474  wcel 1976  cun 3537  c0 3873   class class class wbr 4577  (class class class)co 6527  0cc0 9792  1c1 9793   + caddc 9795  cle 9931  cn 10867  0cn0 11139  cz 11210  ...cfz 12152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153
This theorem is referenced by:  eldioph2lem1  36137
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