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Mirrors > Home > MPE Home > Th. List > fzo0to3tp | Structured version Visualization version 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 11699 | . . 3 ⊢ 3 ∈ ℤ | |
2 | fzoval 12725 | . . 3 ⊢ (3 ∈ ℤ → (0..^3) = (0...(3 − 1))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (0..^3) = (0...(3 − 1)) |
4 | 3m1e2 11447 | . . . 4 ⊢ (3 − 1) = 2 | |
5 | 2cn 11387 | . . . . 5 ⊢ 2 ∈ ℂ | |
6 | 5 | addid2i 10515 | . . . 4 ⊢ (0 + 2) = 2 |
7 | 4, 6 | eqtr4i 2825 | . . 3 ⊢ (3 − 1) = (0 + 2) |
8 | 7 | oveq2i 6890 | . 2 ⊢ (0...(3 − 1)) = (0...(0 + 2)) |
9 | 0z 11676 | . . 3 ⊢ 0 ∈ ℤ | |
10 | fztp 12650 | . . . 4 ⊢ (0 ∈ ℤ → (0...(0 + 2)) = {0, (0 + 1), (0 + 2)}) | |
11 | eqidd 2801 | . . . . 5 ⊢ (0 ∈ ℤ → 0 = 0) | |
12 | 0p1e1 11441 | . . . . . 6 ⊢ (0 + 1) = 1 | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (0 ∈ ℤ → (0 + 1) = 1) |
14 | 6 | a1i 11 | . . . . 5 ⊢ (0 ∈ ℤ → (0 + 2) = 2) |
15 | 11, 13, 14 | tpeq123d 4473 | . . . 4 ⊢ (0 ∈ ℤ → {0, (0 + 1), (0 + 2)} = {0, 1, 2}) |
16 | 10, 15 | eqtrd 2834 | . . 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 2826 | 1 ⊢ (0..^3) = {0, 1, 2} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 {ctp 4373 (class class class)co 6879 0cc0 10225 1c1 10226 + caddc 10228 − cmin 10557 2c2 11367 3c3 11368 ℤcz 11665 ...cfz 12579 ..^cfzo 12719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-om 7301 df-1st 7402 df-2nd 7403 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-er 7983 df-en 8197 df-dom 8198 df-sdom 8199 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-nn 11314 df-2 11375 df-3 11376 df-n0 11580 df-z 11666 df-uz 11930 df-fz 12580 df-fzo 12720 |
This theorem is referenced by: s3fn 13995 wrd3tpop 14032 eqwrds3 14046 wrdl3s3 14047 trgcgrg 25765 tgcgr4 25781 2pthdlem1 27218 wwlks2onv 27241 elwwlks2ons3im 27242 elwwlks2ons3OLD 27244 umgrwwlks2on 27246 3wlkdlem2 27503 upgr3v3e3cycl 27523 prodfzo03 31200 circlevma 31239 circlemethhgt 31240 hgt750lemg 31251 hgt750lemb 31253 hgt750lema 31254 hgt750leme 31255 tgoldbachgtde 31257 tgoldbachgt 31260 |
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