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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 9295 | . . 3 ⊢ 3 ∈ ℤ | |
2 | fzoval 10161 | . . 3 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (0..^3) = (0...(3 − 1)) |
4 | 3m1e2 9052 | . . . 4 ⊢ (3 − 1) = 2 | |
5 | 2cn 9003 | . . . . 5 ⊢ 2 ∈ ℂ | |
6 | 5 | addid2i 8113 | . . . 4 ⊢ (0 + 2) = 2 |
7 | 4, 6 | eqtr4i 2211 | . . 3 ⊢ (3 − 1) = (0 + 2) |
8 | 7 | oveq2i 5899 | . 2 ⊢ (0...(3 − 1)) = (0...(0 + 2)) |
9 | 0z 9277 | . . 3 ⊢ 0 ∈ ℤ | |
10 | fztp 10091 | . . . 4 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}) | |
11 | eqidd 2188 | . . . . 5 ⊢ (0 ∈ ℤ → 0 = 0) | |
12 | 0p1e1 9046 | . . . . . 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 3696 | . . . 4 ⊢ (0 ∈ ℤ → {0, (0 + 1), (0 + 2)} = {0, 1, 2}) |
16 | 10, 15 | eqtrd 2220 | . . 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 2212 | 1 ⊢ (0..^3) = {0, 1, 2} |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 ∈ wcel 2158 {ctp 3606 (class class class)co 5888 0cc0 7824 1c1 7825 + caddc 7827 − cmin 8141 2c2 8983 3c3 8984 ℤcz 9266 ...cfz 10021 ..^cfzo 10155 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-tp 3612 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-2 8991 df-3 8992 df-n0 9190 df-z 9267 df-uz 9542 df-fz 10022 df-fzo 10156 |
This theorem is referenced by: (None) |
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