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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 9291 | . . . . . 6 | |
2 | 1 | zcnd 9132 | . . . . 5 |
3 | 2 | adantl 275 | . . . 4 |
4 | ax-1cn 7681 | . . . 4 | |
5 | npcan 7939 | . . . . 5 | |
6 | 5 | eqcomd 2123 | . . . 4 |
7 | 3, 4, 6 | sylancl 409 | . . 3 |
8 | 7 | oveq2d 5758 | . 2 ..^ ..^ |
9 | eluzp1m1 9305 | . . . 4 | |
10 | 1 | adantl 275 | . . . . 5 |
11 | peano2zm 9050 | . . . . 5 | |
12 | uzid 9296 | . . . . 5 | |
13 | peano2uz 9334 | . . . . 5 | |
14 | 10, 11, 12, 13 | 4syl 18 | . . . 4 |
15 | elfzuzb 9755 | . . . 4 | |
16 | 9, 14, 15 | sylanbrc 413 | . . 3 |
17 | fzosplit 9909 | . . 3 ..^ ..^ ..^ | |
18 | 16, 17 | syl 14 | . 2 ..^ ..^ ..^ |
19 | 1, 11 | syl 14 | . . . . 5 |
20 | 19 | adantl 275 | . . . 4 |
21 | fzosn 9937 | . . . 4 ..^ | |
22 | 20, 21 | syl 14 | . . 3 ..^ |
23 | 22 | uneq2d 3200 | . 2 ..^ ..^ ..^ |
24 | 8, 18, 23 | 3eqtrd 2154 | 1 ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 cun 3039 csn 3497 cfv 5093 (class class class)co 5742 cc 7586 c1 7589 caddc 7591 cmin 7901 cz 9012 cuz 9282 cfz 9745 ..^cfzo 9874 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 df-uz 9283 df-fz 9746 df-fzo 9875 |
This theorem is referenced by: elfzonlteqm1 9942 |
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