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Mirrors > Home > ILE Home > Th. List > fz0fzelfz0 | Unicode version |
Description: If a member of a finite set of sequential integers with a lower bound being a member of a finite set of sequential nonnegative integers with the same upper bound, this member is also a member of the finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 21-Apr-2018.) |
Ref | Expression |
---|---|
fz0fzelfz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2nn0 9128 | . . . 4 | |
2 | elfz2 9036 | . . . . . 6 | |
3 | simplr 496 | . . . . . . . . . . . . . . . . 17 | |
4 | 0red 7120 | . . . . . . . . . . . . . . . . . . . 20 | |
5 | nn0re 8297 | . . . . . . . . . . . . . . . . . . . . 21 | |
6 | 5 | adantr 270 | . . . . . . . . . . . . . . . . . . . 20 |
7 | zre 8355 | . . . . . . . . . . . . . . . . . . . . 21 | |
8 | 7 | adantl 271 | . . . . . . . . . . . . . . . . . . . 20 |
9 | 4, 6, 8 | 3jca 1118 | . . . . . . . . . . . . . . . . . . 19 |
10 | 9 | adantr 270 | . . . . . . . . . . . . . . . . . 18 |
11 | nn0ge0 8313 | . . . . . . . . . . . . . . . . . . . 20 | |
12 | 11 | adantr 270 | . . . . . . . . . . . . . . . . . . 19 |
13 | 12 | anim1i 333 | . . . . . . . . . . . . . . . . . 18 |
14 | letr 7194 | . . . . . . . . . . . . . . . . . 18 | |
15 | 10, 13, 14 | sylc 61 | . . . . . . . . . . . . . . . . 17 |
16 | elnn0z 8364 | . . . . . . . . . . . . . . . . 17 | |
17 | 3, 15, 16 | sylanbrc 408 | . . . . . . . . . . . . . . . 16 |
18 | 17 | exp31 356 | . . . . . . . . . . . . . . 15 |
19 | 18 | com23 77 | . . . . . . . . . . . . . 14 |
20 | 19 | 3ad2ant1 959 | . . . . . . . . . . . . 13 |
21 | 20 | com13 79 | . . . . . . . . . . . 12 |
22 | 21 | adantrd 273 | . . . . . . . . . . 11 |
23 | 22 | 3ad2ant3 961 | . . . . . . . . . 10 |
24 | 23 | imp 122 | . . . . . . . . 9 |
25 | 24 | imp 122 | . . . . . . . 8 |
26 | simpr2 945 | . . . . . . . 8 | |
27 | simplrr 502 | . . . . . . . 8 | |
28 | 25, 26, 27 | 3jca 1118 | . . . . . . 7 |
29 | 28 | ex 113 | . . . . . 6 |
30 | 2, 29 | sylbi 119 | . . . . 5 |
31 | 30 | com12 30 | . . . 4 |
32 | 1, 31 | sylbi 119 | . . 3 |
33 | 32 | imp 122 | . 2 |
34 | elfz2nn0 9128 | . 2 | |
35 | 33, 34 | sylibr 132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 cle 7154 cn0 8288 cz 8351 cfz 9029 |
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-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-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-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-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 |
This theorem is referenced by: fz0fzdiffz0 9141 |
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