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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 9107 | . . 3 ⊢ 3 ∈ ℤ | |
2 | fzoval 9956 | . . 3 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (0..^3) = (0...(3 − 1)) |
4 | 3m1e2 8864 | . . . 4 ⊢ (3 − 1) = 2 | |
5 | 2cn 8815 | . . . . 5 ⊢ 2 ∈ ℂ | |
6 | 5 | addid2i 7929 | . . . 4 ⊢ (0 + 2) = 2 |
7 | 4, 6 | eqtr4i 2164 | . . 3 ⊢ (3 − 1) = (0 + 2) |
8 | 7 | oveq2i 5793 | . 2 ⊢ (0...(3 − 1)) = (0...(0 + 2)) |
9 | 0z 9089 | . . 3 ⊢ 0 ∈ ℤ | |
10 | fztp 9889 | . . . 4 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}) | |
11 | eqidd 2141 | . . . . 5 ⊢ (0 ∈ ℤ → 0 = 0) | |
12 | 0p1e1 8858 | . . . . . 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 3623 | . . . 4 ⊢ (0 ∈ ℤ → {0, (0 + 1), (0 + 2)} = {0, 1, 2}) |
16 | 10, 15 | eqtrd 2173 | . . 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 2165 | 1 ⊢ (0..^3) = {0, 1, 2} |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 {ctp 3534 (class class class)co 5782 0cc0 7644 1c1 7645 + caddc 7647 − cmin 7957 2c2 8795 3c3 8796 ℤcz 9078 ...cfz 9821 ..^cfzo 9950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-tp 3540 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-3 8804 df-n0 9002 df-z 9079 df-uz 9351 df-fz 9822 df-fzo 9951 |
This theorem is referenced by: (None) |
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