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Mirrors > Home > ILE Home > Th. List > fzo0to42pr | GIF version |
Description: A half-open integer range from 0 to 4 is a union of two unordered pairs. (Contributed by Alexander van der Vekens, 17-Nov-2017.) |
Ref | Expression |
---|---|
fzo0to42pr | ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 8198 | . . . 4 ⊢ 2 ∈ ℕ0 | |
2 | 4nn0 8200 | . . . 4 ⊢ 4 ∈ ℕ0 | |
3 | 2re 7985 | . . . . 5 ⊢ 2 ∈ ℝ | |
4 | 4re 7992 | . . . . 5 ⊢ 4 ∈ ℝ | |
5 | 2lt4 8090 | . . . . 5 ⊢ 2 < 4 | |
6 | 3, 4, 5 | ltleii 7120 | . . . 4 ⊢ 2 ≤ 4 |
7 | elfz2nn0 8973 | . . . 4 ⊢ (2 ∈ (0...4) ↔ (2 ∈ ℕ0 ∧ 4 ∈ ℕ0 ∧ 2 ≤ 4)) | |
8 | 1, 2, 6, 7 | mpbir3an 1086 | . . 3 ⊢ 2 ∈ (0...4) |
9 | fzosplit 9033 | . . 3 ⊢ (2 ∈ (0...4) → (0..^4) = ((0..^2) ∪ (2..^4))) | |
10 | 8, 9 | ax-mp 7 | . 2 ⊢ (0..^4) = ((0..^2) ∪ (2..^4)) |
11 | fzo0to2pr 9074 | . . 3 ⊢ (0..^2) = {0, 1} | |
12 | 4z 8275 | . . . . 5 ⊢ 4 ∈ ℤ | |
13 | fzoval 9005 | . . . . 5 ⊢ (4 ∈ ℤ → (2..^4) = (2...(4 − 1))) | |
14 | 12, 13 | ax-mp 7 | . . . 4 ⊢ (2..^4) = (2...(4 − 1)) |
15 | 4cn 7993 | . . . . . . . 8 ⊢ 4 ∈ ℂ | |
16 | ax-1cn 6977 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
17 | 3cn 7990 | . . . . . . . 8 ⊢ 3 ∈ ℂ | |
18 | df-4 7975 | . . . . . . . . . 10 ⊢ 4 = (3 + 1) | |
19 | 17, 16 | addcomi 7157 | . . . . . . . . . 10 ⊢ (3 + 1) = (1 + 3) |
20 | 18, 19 | eqtri 2060 | . . . . . . . . 9 ⊢ 4 = (1 + 3) |
21 | 20 | eqcomi 2044 | . . . . . . . 8 ⊢ (1 + 3) = 4 |
22 | 15, 16, 17, 21 | subaddrii 7300 | . . . . . . 7 ⊢ (4 − 1) = 3 |
23 | df-3 7974 | . . . . . . 7 ⊢ 3 = (2 + 1) | |
24 | 22, 23 | eqtri 2060 | . . . . . 6 ⊢ (4 − 1) = (2 + 1) |
25 | 24 | oveq2i 5523 | . . . . 5 ⊢ (2...(4 − 1)) = (2...(2 + 1)) |
26 | 2z 8273 | . . . . . 6 ⊢ 2 ∈ ℤ | |
27 | fzpr 8939 | . . . . . 6 ⊢ (2 ∈ ℤ → (2...(2 + 1)) = {2, (2 + 1)}) | |
28 | 26, 27 | ax-mp 7 | . . . . 5 ⊢ (2...(2 + 1)) = {2, (2 + 1)} |
29 | 25, 28 | eqtri 2060 | . . . 4 ⊢ (2...(4 − 1)) = {2, (2 + 1)} |
30 | 23 | eqcomi 2044 | . . . . 5 ⊢ (2 + 1) = 3 |
31 | 30 | preq2i 3451 | . . . 4 ⊢ {2, (2 + 1)} = {2, 3} |
32 | 14, 29, 31 | 3eqtri 2064 | . . 3 ⊢ (2..^4) = {2, 3} |
33 | 11, 32 | uneq12i 3095 | . 2 ⊢ ((0..^2) ∪ (2..^4)) = ({0, 1} ∪ {2, 3}) |
34 | 10, 33 | eqtri 2060 | 1 ⊢ (0..^4) = ({0, 1} ∪ {2, 3}) |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 ∪ cun 2915 {cpr 3376 class class class wbr 3764 (class class class)co 5512 0cc0 6889 1c1 6890 + caddc 6892 ≤ cle 7061 − cmin 7182 2c2 7964 3c3 7965 4c4 7966 ℕ0cn0 8181 ℤcz 8245 ...cfz 8874 ..^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-2 7973 df-3 7974 df-4 7975 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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