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Mirrors > Home > ILE Home > Th. List > fzo0to3tp | GIF version |
Description: A half-open integer range from 0 to 3 is an unordered triple. (Contributed by Alexander van der Vekens, 9-Nov-2017.) |
Ref | Expression |
---|---|
fzo0to3tp | ⊢ (0..^3) = {0, 1, 2} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3z 9301 | . . 3 ⊢ 3 ∈ ℤ | |
2 | fzoval 10167 | . . 3 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (0..^3) = (0...(3 − 1)) |
4 | 3m1e2 9058 | . . . 4 ⊢ (3 − 1) = 2 | |
5 | 2cn 9009 | . . . . 5 ⊢ 2 ∈ ℂ | |
6 | 5 | addid2i 8119 | . . . 4 ⊢ (0 + 2) = 2 |
7 | 4, 6 | eqtr4i 2213 | . . 3 ⊢ (3 − 1) = (0 + 2) |
8 | 7 | oveq2i 5902 | . 2 ⊢ (0...(3 − 1)) = (0...(0 + 2)) |
9 | 0z 9283 | . . 3 ⊢ 0 ∈ ℤ | |
10 | fztp 10097 | . . . 4 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}) | |
11 | eqidd 2190 | . . . . 5 ⊢ (0 ∈ ℤ → 0 = 0) | |
12 | 0p1e1 9052 | . . . . . 6 ⊢ (0 + 1) = 1 | |
13 | 12 | a1i 9 | . . . . 5 ⊢ (0 ∈ ℤ → (0 + 1) = 1) |
14 | 6 | a1i 9 | . . . . 5 ⊢ (0 ∈ ℤ → (0 + 2) = 2) |
15 | 11, 13, 14 | tpeq123d 3699 | . . . 4 ⊢ (0 ∈ ℤ → {0, (0 + 1), (0 + 2)} = {0, 1, 2}) |
16 | 10, 15 | eqtrd 2222 | . . 3 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, 1, 2}) |
17 | 9, 16 | ax-mp 5 | . 2 ⊢ (0...(0 + 2)) = {0, 1, 2} |
18 | 3, 8, 17 | 3eqtri 2214 | 1 ⊢ (0..^3) = {0, 1, 2} |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 {ctp 3609 (class class class)co 5891 0cc0 7830 1c1 7831 + caddc 7833 − cmin 8147 2c2 8989 3c3 8990 ℤcz 9272 ...cfz 10027 ..^cfzo 10161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-addcom 7930 ax-addass 7932 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-0id 7938 ax-rnegex 7939 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-inn 8939 df-2 8997 df-3 8998 df-n0 9196 df-z 9273 df-uz 9548 df-fz 10028 df-fzo 10162 |
This theorem is referenced by: (None) |
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