Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7781 | . . 3 | |
2 | elfzelz 9806 | . . . . . 6 | |
3 | 2 | zred 9173 | . . . . 5 |
4 | 3 | adantl 275 | . . . 4 |
5 | peano2re 7898 | . . . 4 | |
6 | 4, 5 | syl 14 | . . 3 |
7 | peano2re 7898 | . . . . 5 | |
8 | 1, 7 | syl 14 | . . . 4 |
9 | 1 | ltp1d 8688 | . . . 4 |
10 | elfzle1 9807 | . . . . . 6 | |
11 | 10 | adantl 275 | . . . . 5 |
12 | 1re 7765 | . . . . . . 7 | |
13 | leadd1 8192 | . . . . . . 7 | |
14 | 12, 12, 13 | mp3an13 1306 | . . . . . 6 |
15 | 4, 14 | syl 14 | . . . . 5 |
16 | 11, 15 | mpbid 146 | . . . 4 |
17 | 1, 8, 6, 9, 16 | ltletrd 8185 | . . 3 |
18 | 1, 6, 17 | ltled 7881 | . 2 |
19 | elfzle2 9808 | . . . 4 | |
20 | 19 | adantl 275 | . . 3 |
21 | nnz 9073 | . . . . . 6 | |
22 | 21 | adantr 274 | . . . . 5 |
23 | 22 | zred 9173 | . . . 4 |
24 | leaddsub 8200 | . . . . 5 | |
25 | 12, 24 | mp3an2 1303 | . . . 4 |
26 | 4, 23, 25 | syl2anc 408 | . . 3 |
27 | 20, 26 | mpbird 166 | . 2 |
28 | 2 | peano2zd 9176 | . . . 4 |
29 | 28 | adantl 275 | . . 3 |
30 | 1z 9080 | . . . 4 | |
31 | elfz 9796 | . . . 4 | |
32 | 30, 31 | mp3an2 1303 | . . 3 |
33 | 29, 22, 32 | syl2anc 408 | . 2 |
34 | 18, 27, 33 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3929 (class class class)co 5774 cr 7619 c1 7621 caddc 7623 cle 7801 cmin 7933 cn 8720 cz 9054 cfz 9790 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |