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Theorem fzsplit1nn0 37843
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 11496 . . 3 (𝐴 ∈ ℕ0 ↔ (𝐴 ∈ ℕ ∨ 𝐴 = 0))
2 nnge1 11248 . . . . . . . 8 (𝐴 ∈ ℕ → 1 ≤ 𝐴)
32adantr 466 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ≤ 𝐴)
4 simprr 756 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴𝐵)
5 nnz 11601 . . . . . . . . 9 (𝐴 ∈ ℕ → 𝐴 ∈ ℤ)
65adantr 466 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ ℤ)
7 1zzd 11610 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 1 ∈ ℤ)
8 nn0z 11602 . . . . . . . . 9 (𝐵 ∈ ℕ0𝐵 ∈ ℤ)
98ad2antrl 707 . . . . . . . 8 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐵 ∈ ℤ)
10 elfz 12539 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
116, 7, 9, 10syl3anc 1476 . . . . . . 7 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 ∈ (1...𝐵) ↔ (1 ≤ 𝐴𝐴𝐵)))
123, 4, 11mpbir2and 692 . . . . . 6 ((𝐴 ∈ ℕ ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → 𝐴 ∈ (1...𝐵))
13 fzsplit 12574 . . . . . 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 3908 . . . . . 6 ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)) = (((𝐴 + 1)...𝐵) ∪ (1...𝐴))
16 oveq1 6800 . . . . . . . . . . 11 (𝐴 = 0 → (𝐴 + 1) = (0 + 1))
1716adantr 466 . . . . . . . . . 10 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = (0 + 1))
18 0p1e1 11334 . . . . . . . . . 10 (0 + 1) = 1
1917, 18syl6eq 2821 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (𝐴 + 1) = 1)
2019oveq1d 6808 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → ((𝐴 + 1)...𝐵) = (1...𝐵))
21 oveq2 6801 . . . . . . . . . 10 (𝐴 = 0 → (1...𝐴) = (1...0))
2221adantr 466 . . . . . . . . 9 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = (1...0))
23 fz10 12569 . . . . . . . . 9 (1...0) = ∅
2422, 23syl6eq 2821 . . . . . . . 8 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐴) = ∅)
2520, 24uneq12d 3919 . . . . . . 7 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = ((1...𝐵) ∪ ∅))
26 un0 4111 . . . . . . 7 ((1...𝐵) ∪ ∅) = (1...𝐵)
2725, 26syl6eq 2821 . . . . . 6 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (((𝐴 + 1)...𝐵) ∪ (1...𝐴)) = (1...𝐵))
2815, 27syl5req 2818 . . . . 5 ((𝐴 = 0 ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
2914, 28jaoian 941 . . . 4 (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ (𝐵 ∈ ℕ0𝐴𝐵)) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
3029ex 397 . . 3 ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
311, 30sylbi 207 . 2 (𝐴 ∈ ℕ0 → ((𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵))))
32313impib 1108 1 ((𝐴 ∈ ℕ0𝐵 ∈ ℕ0𝐴𝐵) → (1...𝐵) = ((1...𝐴) ∪ ((𝐴 + 1)...𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  cun 3721  c0 4063   class class class wbr 4786  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141  cle 10277  cn 11222  0cn0 11494  cz 11579  ...cfz 12533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534
This theorem is referenced by:  eldioph2lem1  37849
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