Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fzsplit2 | Structured version Visualization version Unicode version |
Description: Split a finite interval of integers into two parts. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
fzsplit2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzelz 12342 | . . . . . . 7 | |
2 | 1 | zred 11482 | . . . . . 6 |
3 | eluzel2 11692 | . . . . . . . 8 | |
4 | 3 | adantl 482 | . . . . . . 7 |
5 | 4 | zred 11482 | . . . . . 6 |
6 | lelttric 10144 | . . . . . 6 | |
7 | 2, 5, 6 | syl2anr 495 | . . . . 5 |
8 | elfzuz 12338 | . . . . . . 7 | |
9 | elfz5 12334 | . . . . . . 7 | |
10 | 8, 4, 9 | syl2anr 495 | . . . . . 6 |
11 | simpl 473 | . . . . . . . . 9 | |
12 | eluzelz 11697 | . . . . . . . . 9 | |
13 | 11, 12 | syl 17 | . . . . . . . 8 |
14 | eluz 11701 | . . . . . . . 8 | |
15 | 13, 1, 14 | syl2an 494 | . . . . . . 7 |
16 | elfzuz3 12339 | . . . . . . . . 9 | |
17 | 16 | adantl 482 | . . . . . . . 8 |
18 | elfzuzb 12336 | . . . . . . . . 9 | |
19 | 18 | rbaib 947 | . . . . . . . 8 |
20 | 17, 19 | syl 17 | . . . . . . 7 |
21 | zltp1le 11427 | . . . . . . . 8 | |
22 | 4, 1, 21 | syl2an 494 | . . . . . . 7 |
23 | 15, 20, 22 | 3bitr4d 300 | . . . . . 6 |
24 | 10, 23 | orbi12d 746 | . . . . 5 |
25 | 7, 24 | mpbird 247 | . . . 4 |
26 | elfzuz 12338 | . . . . . . 7 | |
27 | 26 | adantl 482 | . . . . . 6 |
28 | simpr 477 | . . . . . . 7 | |
29 | elfzuz3 12339 | . . . . . . 7 | |
30 | uztrn 11704 | . . . . . . 7 | |
31 | 28, 29, 30 | syl2an 494 | . . . . . 6 |
32 | elfzuzb 12336 | . . . . . 6 | |
33 | 27, 31, 32 | sylanbrc 698 | . . . . 5 |
34 | elfzuz 12338 | . . . . . . 7 | |
35 | uztrn 11704 | . . . . . . 7 | |
36 | 34, 11, 35 | syl2anr 495 | . . . . . 6 |
37 | elfzuz3 12339 | . . . . . . 7 | |
38 | 37 | adantl 482 | . . . . . 6 |
39 | 36, 38, 32 | sylanbrc 698 | . . . . 5 |
40 | 33, 39 | jaodan 826 | . . . 4 |
41 | 25, 40 | impbida 877 | . . 3 |
42 | elun 3753 | . . 3 | |
43 | 41, 42 | syl6bbr 278 | . 2 |
44 | 43 | eqrdv 2620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cun 3572 class class class wbr 4653 cfv 5888 (class class class)co 6650 cr 9935 c1 9937 caddc 9939 clt 10074 cle 10075 cz 11377 cuz 11687 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: fzsplit 12367 fzpred 12389 fz0to4untppr 12442 fallfacval4 14774 fsumharmonic 24738 gausslemma2dlem6 25097 dchrisum0lem1b 25204 dchrisum0lem1 25205 dchrisum0lem3 25208 pntrsumbnd2 25256 pntrlog2bndlem6a 25271 pntlemf 25294 fzspl 29550 poimirlem1 33410 poimirlem2 33411 poimirlem3 33412 poimirlem4 33413 poimirlem6 33415 poimirlem7 33416 poimirlem8 33417 poimirlem12 33421 poimirlem13 33422 poimirlem14 33423 poimirlem16 33425 poimirlem17 33426 poimirlem18 33427 poimirlem19 33428 poimirlem20 33429 poimirlem21 33430 poimirlem22 33431 poimirlem23 33432 poimirlem24 33433 poimirlem25 33434 poimirlem26 33435 poimirlem28 33437 poimirlem29 33438 poimirlem31 33440 poimirlem32 33441 |
Copyright terms: Public domain | W3C validator |