Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > elfz0fzfz0 | Structured version Visualization version Unicode version |
Description: A member of a finite set of sequential nonnegative integers is a member of a finite set of sequential nonnegative integers with a member of a finite set of sequential nonnegative integers starting at the upper bound of the first interval. (Contributed by Alexander van der Vekens, 27-May-2018.) |
Ref | Expression |
---|---|
elfz0fzfz0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2nn0 12431 | . . . 4 | |
2 | elfz2 12333 | . . . . . 6 | |
3 | nn0re 11301 | . . . . . . . . . . . . . . . . . 18 | |
4 | nn0re 11301 | . . . . . . . . . . . . . . . . . 18 | |
5 | zre 11381 | . . . . . . . . . . . . . . . . . 18 | |
6 | 3, 4, 5 | 3anim123i 1247 | . . . . . . . . . . . . . . . . 17 |
7 | 6 | 3expa 1265 | . . . . . . . . . . . . . . . 16 |
8 | letr 10131 | . . . . . . . . . . . . . . . 16 | |
9 | 7, 8 | syl 17 | . . . . . . . . . . . . . . 15 |
10 | simplll 798 | . . . . . . . . . . . . . . . . 17 | |
11 | simpr 477 | . . . . . . . . . . . . . . . . . . 19 | |
12 | 11 | adantr 481 | . . . . . . . . . . . . . . . . . 18 |
13 | elnn0z 11390 | . . . . . . . . . . . . . . . . . . . . . 22 | |
14 | 0red 10041 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
15 | zre 11381 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27 | |
16 | 15 | adantr 481 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
17 | 5 | adantl 482 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 |
18 | letr 10131 | . . . . . . . . . . . . . . . . . . . . . . . . . 26 | |
19 | 14, 16, 17, 18 | syl3anc 1326 | . . . . . . . . . . . . . . . . . . . . . . . . 25 |
20 | 19 | exp4b 632 | . . . . . . . . . . . . . . . . . . . . . . . 24 |
21 | 20 | com23 86 | . . . . . . . . . . . . . . . . . . . . . . 23 |
22 | 21 | imp 445 | . . . . . . . . . . . . . . . . . . . . . 22 |
23 | 13, 22 | sylbi 207 | . . . . . . . . . . . . . . . . . . . . 21 |
24 | 23 | adantr 481 | . . . . . . . . . . . . . . . . . . . 20 |
25 | 24 | imp 445 | . . . . . . . . . . . . . . . . . . 19 |
26 | 25 | imp 445 | . . . . . . . . . . . . . . . . . 18 |
27 | elnn0z 11390 | . . . . . . . . . . . . . . . . . 18 | |
28 | 12, 26, 27 | sylanbrc 698 | . . . . . . . . . . . . . . . . 17 |
29 | simpr 477 | . . . . . . . . . . . . . . . . 17 | |
30 | 10, 28, 29 | 3jca 1242 | . . . . . . . . . . . . . . . 16 |
31 | 30 | ex 450 | . . . . . . . . . . . . . . 15 |
32 | 9, 31 | syld 47 | . . . . . . . . . . . . . 14 |
33 | 32 | exp4b 632 | . . . . . . . . . . . . 13 |
34 | 33 | com23 86 | . . . . . . . . . . . 12 |
35 | 34 | 3impia 1261 | . . . . . . . . . . 11 |
36 | 35 | com13 88 | . . . . . . . . . 10 |
37 | 36 | adantr 481 | . . . . . . . . 9 |
38 | 37 | com12 32 | . . . . . . . 8 |
39 | 38 | 3ad2ant3 1084 | . . . . . . 7 |
40 | 39 | imp 445 | . . . . . 6 |
41 | 2, 40 | sylbi 207 | . . . . 5 |
42 | 41 | com12 32 | . . . 4 |
43 | 1, 42 | sylbi 207 | . . 3 |
44 | 43 | imp 445 | . 2 |
45 | elfz2nn0 12431 | . 2 | |
46 | 44, 45 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wcel 1990 class class class wbr 4653 (class class class)co 6650 cr 9935 cc0 9936 cle 10075 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: swrdccatin12lem2c 13488 swrdccatin12 13491 pfxccatin12 41425 |
Copyright terms: Public domain | W3C validator |