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Mirrors > Home > ILE Home > Th. List > fzosplitprm1 | Unicode version |
Description: Extending a half-open integer range by an unordered pair at the end. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
Ref | Expression |
---|---|
fzosplitprm1 | ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 966 | . . . 4 | |
2 | simp2 967 | . . . 4 | |
3 | zre 9026 | . . . . . 6 | |
4 | zre 9026 | . . . . . 6 | |
5 | ltle 7819 | . . . . . 6 | |
6 | 3, 4, 5 | syl2an 287 | . . . . 5 |
7 | 6 | 3impia 1163 | . . . 4 |
8 | eluz2 9300 | . . . 4 | |
9 | 1, 2, 7, 8 | syl3anbrc 1150 | . . 3 |
10 | fzosplitsn 9978 | . . 3 ..^ ..^ | |
11 | 9, 10 | syl 14 | . 2 ..^ ..^ |
12 | zcn 9027 | . . . . . . 7 | |
13 | ax-1cn 7681 | . . . . . . 7 | |
14 | npcan 7939 | . . . . . . . 8 | |
15 | 14 | eqcomd 2123 | . . . . . . 7 |
16 | 12, 13, 15 | sylancl 409 | . . . . . 6 |
17 | 16 | 3ad2ant2 988 | . . . . 5 |
18 | 17 | oveq2d 5758 | . . . 4 ..^ ..^ |
19 | peano2zm 9060 | . . . . . . 7 | |
20 | 19 | 3ad2ant2 988 | . . . . . 6 |
21 | zltlem1 9079 | . . . . . . 7 | |
22 | 21 | biimp3a 1308 | . . . . . 6 |
23 | eluz2 9300 | . . . . . 6 | |
24 | 1, 20, 22, 23 | syl3anbrc 1150 | . . . . 5 |
25 | fzosplitsn 9978 | . . . . 5 ..^ ..^ | |
26 | 24, 25 | syl 14 | . . . 4 ..^ ..^ |
27 | 18, 26 | eqtrd 2150 | . . 3 ..^ ..^ |
28 | 27 | uneq1d 3199 | . 2 ..^ ..^ |
29 | unass 3203 | . . 3 ..^ ..^ | |
30 | df-pr 3504 | . . . . . 6 | |
31 | 30 | eqcomi 2121 | . . . . 5 |
32 | 31 | a1i 9 | . . . 4 |
33 | 32 | uneq2d 3200 | . . 3 ..^ ..^ |
34 | 29, 33 | syl5eq 2162 | . 2 ..^ ..^ |
35 | 11, 28, 34 | 3eqtrd 2154 | 1 ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wcel 1465 cun 3039 csn 3497 cpr 3498 class class class wbr 3899 cfv 5093 (class class class)co 5742 cc 7586 cr 7587 c1 7589 caddc 7591 clt 7768 cle 7769 cmin 7901 cz 9022 cuz 9294 ..^cfzo 9887 |
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 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 df-fzo 9888 |
This theorem is referenced by: (None) |
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