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Mirrors > Home > MPE Home > Th. List > difelfzle | Structured version Visualization version Unicode version |
Description: The difference of two integers from a finite set of sequential nonnegative integers is also element of this finite set of sequential integers. (Contributed by Alexander van der Vekens, 12-Jun-2018.) |
Ref | Expression |
---|---|
difelfzle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfznn0 12433 | . . . . 5 | |
2 | elfznn0 12433 | . . . . 5 | |
3 | nn0z 11400 | . . . . . . . . 9 | |
4 | nn0z 11400 | . . . . . . . . 9 | |
5 | zsubcl 11419 | . . . . . . . . 9 | |
6 | 3, 4, 5 | syl2anr 495 | . . . . . . . 8 |
7 | 6 | adantr 481 | . . . . . . 7 |
8 | nn0re 11301 | . . . . . . . . 9 | |
9 | nn0re 11301 | . . . . . . . . 9 | |
10 | subge0 10541 | . . . . . . . . 9 | |
11 | 8, 9, 10 | syl2anr 495 | . . . . . . . 8 |
12 | 11 | biimpar 502 | . . . . . . 7 |
13 | 7, 12 | jca 554 | . . . . . 6 |
14 | 13 | exp31 630 | . . . . 5 |
15 | 1, 2, 14 | syl2im 40 | . . . 4 |
16 | 15 | 3imp 1256 | . . 3 |
17 | elnn0z 11390 | . . 3 | |
18 | 16, 17 | sylibr 224 | . 2 |
19 | elfz3nn0 12434 | . . 3 | |
20 | 19 | 3ad2ant1 1082 | . 2 |
21 | elfz2nn0 12431 | . . . . . 6 | |
22 | 8 | 3ad2ant1 1082 | . . . . . . . . 9 |
23 | resubcl 10345 | . . . . . . . . 9 | |
24 | 22, 9, 23 | syl2an 494 | . . . . . . . 8 |
25 | 22 | adantr 481 | . . . . . . . 8 |
26 | nn0re 11301 | . . . . . . . . . 10 | |
27 | 26 | 3ad2ant2 1083 | . . . . . . . . 9 |
28 | 27 | adantr 481 | . . . . . . . 8 |
29 | nn0ge0 11318 | . . . . . . . . . 10 | |
30 | 29 | adantl 482 | . . . . . . . . 9 |
31 | subge02 10544 | . . . . . . . . . 10 | |
32 | 22, 9, 31 | syl2an 494 | . . . . . . . . 9 |
33 | 30, 32 | mpbid 222 | . . . . . . . 8 |
34 | simpl3 1066 | . . . . . . . 8 | |
35 | 24, 25, 28, 33, 34 | letrd 10194 | . . . . . . 7 |
36 | 35 | ex 450 | . . . . . 6 |
37 | 21, 36 | sylbi 207 | . . . . 5 |
38 | 1, 37 | syl5com 31 | . . . 4 |
39 | 38 | a1dd 50 | . . 3 |
40 | 39 | 3imp 1256 | . 2 |
41 | elfz2nn0 12431 | . 2 | |
42 | 18, 20, 40, 41 | syl3anbrc 1246 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 cc0 9936 cle 10075 cmin 10266 cn0 11292 cz 11377 cfz 12326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: 2cshwcshw 13571 |
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