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Mirrors > Home > ILE Home > Th. List > fznatpl1 | Unicode version |
Description: Shift membership in a finite sequence of naturals. (Contributed by Scott Fenton, 17-Jul-2013.) |
Ref | Expression |
---|---|
fznatpl1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 7134 | . . 3 | |
2 | elfzelz 9045 | . . . . . 6 | |
3 | 2 | zred 8469 | . . . . 5 |
4 | 3 | adantl 271 | . . . 4 |
5 | peano2re 7244 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | peano2re 7244 | . . . . 5 | |
8 | 1, 7 | syl 14 | . . . 4 |
9 | 1 | ltp1d 8008 | . . . 4 |
10 | elfzle1 9046 | . . . . . 6 | |
11 | 10 | adantl 271 | . . . . 5 |
12 | 1re 7118 | . . . . . . 7 | |
13 | leadd1 7534 | . . . . . . 7 | |
14 | 12, 12, 13 | mp3an13 1259 | . . . . . 6 |
15 | 4, 14 | syl 14 | . . . . 5 |
16 | 11, 15 | mpbid 145 | . . . 4 |
17 | 1, 8, 6, 9, 16 | ltletrd 7527 | . . 3 |
18 | 1, 6, 17 | ltled 7228 | . 2 |
19 | elfzle2 9047 | . . . 4 | |
20 | 19 | adantl 271 | . . 3 |
21 | nnz 8370 | . . . . . 6 | |
22 | 21 | adantr 270 | . . . . 5 |
23 | 22 | zred 8469 | . . . 4 |
24 | leaddsub 7542 | . . . . 5 | |
25 | 12, 24 | mp3an2 1256 | . . . 4 |
26 | 4, 23, 25 | syl2anc 403 | . . 3 |
27 | 20, 26 | mpbird 165 | . 2 |
28 | 2 | peano2zd 8472 | . . . 4 |
29 | 28 | adantl 271 | . . 3 |
30 | 1z 8377 | . . . 4 | |
31 | elfz 9035 | . . . 4 | |
32 | 30, 31 | mp3an2 1256 | . . 3 |
33 | 29, 22, 32 | syl2anc 403 | . 2 |
34 | 18, 27, 33 | mpbir2and 885 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 c1 6982 caddc 6984 cle 7154 cmin 7279 cn 8039 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: (None) |
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