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Mirrors > Home > ILE Home > Th. List > fzosplitprm1 | GIF version |
Description: Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
Ref | Expression |
---|---|
fzosplitprm1 | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 904 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℤ) | |
2 | simp2 905 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℤ) | |
3 | zre 8249 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℝ) | |
4 | zre 8249 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℝ) | |
5 | ltle 7105 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
6 | 3, 4, 5 | syl2an 273 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
7 | 6 | 3impia 1101 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐵) |
8 | eluz2 8479 | . . . 4 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ≤ 𝐵)) | |
9 | 1, 2, 7, 8 | syl3anbrc 1088 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐵 ∈ (ℤ≥‘𝐴)) |
10 | fzosplitsn 9089 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘𝐴) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵})) | |
11 | 9, 10 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^𝐵) ∪ {𝐵})) |
12 | zcn 8250 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℂ) | |
13 | ax-1cn 6977 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
14 | npcan 7220 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝐵 − 1) + 1) = 𝐵) | |
15 | 14 | eqcomd 2045 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 1 ∈ ℂ) → 𝐵 = ((𝐵 − 1) + 1)) |
16 | 12, 13, 15 | sylancl 392 | . . . . . 6 ⊢ (𝐵 ∈ ℤ → 𝐵 = ((𝐵 − 1) + 1)) |
17 | 16 | 3ad2ant2 926 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐵 = ((𝐵 − 1) + 1)) |
18 | 17 | oveq2d 5528 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^𝐵) = (𝐴..^((𝐵 − 1) + 1))) |
19 | peano2zm 8283 | . . . . . . 7 ⊢ (𝐵 ∈ ℤ → (𝐵 − 1) ∈ ℤ) | |
20 | 19 | 3ad2ant2 926 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 − 1) ∈ ℤ) |
21 | zltlem1 8301 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 < 𝐵 ↔ 𝐴 ≤ (𝐵 − 1))) | |
22 | 21 | biimp3a 1235 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → 𝐴 ≤ (𝐵 − 1)) |
23 | eluz2 8479 | . . . . . 6 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ (𝐵 − 1) ∈ ℤ ∧ 𝐴 ≤ (𝐵 − 1))) | |
24 | 1, 20, 22, 23 | syl3anbrc 1088 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐵 − 1) ∈ (ℤ≥‘𝐴)) |
25 | fzosplitsn 9089 | . . . . 5 ⊢ ((𝐵 − 1) ∈ (ℤ≥‘𝐴) → (𝐴..^((𝐵 − 1) + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) | |
26 | 24, 25 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^((𝐵 − 1) + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
27 | 18, 26 | eqtrd 2072 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^𝐵) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)})) |
28 | 27 | uneq1d 3096 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^𝐵) ∪ {𝐵}) = (((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) ∪ {𝐵})) |
29 | unass 3100 | . . 3 ⊢ (((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) ∪ {𝐵}) = ((𝐴..^(𝐵 − 1)) ∪ ({(𝐵 − 1)} ∪ {𝐵})) | |
30 | df-pr 3382 | . . . . . 6 ⊢ {(𝐵 − 1), 𝐵} = ({(𝐵 − 1)} ∪ {𝐵}) | |
31 | 30 | eqcomi 2044 | . . . . 5 ⊢ ({(𝐵 − 1)} ∪ {𝐵}) = {(𝐵 − 1), 𝐵} |
32 | 31 | a1i 9 | . . . 4 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ({(𝐵 − 1)} ∪ {𝐵}) = {(𝐵 − 1), 𝐵}) |
33 | 32 | uneq2d 3097 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → ((𝐴..^(𝐵 − 1)) ∪ ({(𝐵 − 1)} ∪ {𝐵})) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
34 | 29, 33 | syl5eq 2084 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1)}) ∪ {𝐵}) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
35 | 11, 28, 34 | 3eqtrd 2076 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 < 𝐵) → (𝐴..^(𝐵 + 1)) = ((𝐴..^(𝐵 − 1)) ∪ {(𝐵 − 1), 𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 ∪ cun 2915 {csn 3375 {cpr 3376 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℂcc 6887 ℝcr 6888 1c1 6890 + caddc 6892 < clt 7060 ≤ cle 7061 − cmin 7182 ℤcz 8245 ℤ≥cuz 8473 ..^cfzo 8999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-apti 6999 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-fz 8875 df-fzo 9000 |
This theorem is referenced by: (None) |
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