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Mirrors > Home > ILE Home > Th. List > fzosplitsnm1 | Unicode version |
Description: Removing a singleton from a half-open integer range at the end. (Contributed by Alexander van der Vekens, 23-Mar-2018.) |
Ref | Expression |
---|---|
fzosplitsnm1 | ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelz 8628 | . . . . . 6 | |
2 | 1 | zcnd 8470 | . . . . 5 |
3 | 2 | adantl 271 | . . . 4 |
4 | ax-1cn 7069 | . . . 4 | |
5 | npcan 7317 | . . . . 5 | |
6 | 5 | eqcomd 2086 | . . . 4 |
7 | 3, 4, 6 | sylancl 404 | . . 3 |
8 | 7 | oveq2d 5548 | . 2 ..^ ..^ |
9 | eluzp1m1 8642 | . . . 4 | |
10 | 1 | adantl 271 | . . . . 5 |
11 | peano2zm 8389 | . . . . 5 | |
12 | uzid 8633 | . . . . 5 | |
13 | peano2uz 8671 | . . . . 5 | |
14 | 10, 11, 12, 13 | 4syl 18 | . . . 4 |
15 | elfzuzb 9039 | . . . 4 | |
16 | 9, 14, 15 | sylanbrc 408 | . . 3 |
17 | fzosplit 9186 | . . 3 ..^ ..^ ..^ | |
18 | 16, 17 | syl 14 | . 2 ..^ ..^ ..^ |
19 | 1, 11 | syl 14 | . . . . 5 |
20 | 19 | adantl 271 | . . . 4 |
21 | fzosn 9214 | . . . 4 ..^ | |
22 | 20, 21 | syl 14 | . . 3 ..^ |
23 | 22 | uneq2d 3126 | . 2 ..^ ..^ ..^ |
24 | 8, 18, 23 | 3eqtrd 2117 | 1 ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 cun 2971 csn 3398 cfv 4922 (class class class)co 5532 cc 6979 c1 6982 caddc 6984 cmin 7279 cz 8351 cuz 8619 cfz 9029 ..^cfzo 9152 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-fz 9030 df-fzo 9153 |
This theorem is referenced by: elfzonlteqm1 9219 |
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