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Mirrors > Home > ILE Home > Th. List > elfzmlbp | Unicode version |
Description: Subtracting the lower bound of a finite set of sequential integers from an element of this set. (Contributed by Alexander van der Vekens, 29-Mar-2018.) |
Ref | Expression |
---|---|
elfzmlbp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfz2 9036 | . . . 4 | |
2 | znn0sub 8416 | . . . . . . . . . . . . . 14 | |
3 | 2 | adantr 270 | . . . . . . . . . . . . 13 |
4 | 3 | biimpcd 157 | . . . . . . . . . . . 12 |
5 | 4 | adantr 270 | . . . . . . . . . . 11 |
6 | 5 | impcom 123 | . . . . . . . . . 10 |
7 | zre 8355 | . . . . . . . . . . . . . . 15 | |
8 | 7 | adantr 270 | . . . . . . . . . . . . . 14 |
9 | 8 | adantr 270 | . . . . . . . . . . . . 13 |
10 | zre 8355 | . . . . . . . . . . . . . . 15 | |
11 | 10 | adantl 271 | . . . . . . . . . . . . . 14 |
12 | 11 | adantr 270 | . . . . . . . . . . . . 13 |
13 | zaddcl 8391 | . . . . . . . . . . . . . . 15 | |
14 | 13 | adantlr 460 | . . . . . . . . . . . . . 14 |
15 | 14 | zred 8469 | . . . . . . . . . . . . 13 |
16 | letr 7194 | . . . . . . . . . . . . 13 | |
17 | 9, 12, 15, 16 | syl3anc 1169 | . . . . . . . . . . . 12 |
18 | zre 8355 | . . . . . . . . . . . . . 14 | |
19 | addge01 7576 | . . . . . . . . . . . . . 14 | |
20 | 8, 18, 19 | syl2an 283 | . . . . . . . . . . . . 13 |
21 | elnn0z 8364 | . . . . . . . . . . . . . . 15 | |
22 | 21 | simplbi2 377 | . . . . . . . . . . . . . 14 |
23 | 22 | adantl 271 | . . . . . . . . . . . . 13 |
24 | 20, 23 | sylbird 168 | . . . . . . . . . . . 12 |
25 | 17, 24 | syld 44 | . . . . . . . . . . 11 |
26 | 25 | imp 122 | . . . . . . . . . 10 |
27 | df-3an 921 | . . . . . . . . . . . . . . . 16 | |
28 | 3ancoma 926 | . . . . . . . . . . . . . . . 16 | |
29 | 27, 28 | bitr3i 184 | . . . . . . . . . . . . . . 15 |
30 | 10, 7, 18 | 3anim123i 1123 | . . . . . . . . . . . . . . 15 |
31 | 29, 30 | sylbi 119 | . . . . . . . . . . . . . 14 |
32 | lesubadd2 7539 | . . . . . . . . . . . . . 14 | |
33 | 31, 32 | syl 14 | . . . . . . . . . . . . 13 |
34 | 33 | biimprcd 158 | . . . . . . . . . . . 12 |
35 | 34 | adantl 271 | . . . . . . . . . . 11 |
36 | 35 | impcom 123 | . . . . . . . . . 10 |
37 | 6, 26, 36 | 3jca 1118 | . . . . . . . . 9 |
38 | 37 | exp31 356 | . . . . . . . 8 |
39 | 38 | com23 77 | . . . . . . 7 |
40 | 39 | 3adant2 957 | . . . . . 6 |
41 | 40 | imp 122 | . . . . 5 |
42 | 41 | com12 30 | . . . 4 |
43 | 1, 42 | syl5bi 150 | . . 3 |
44 | 43 | imp 122 | . 2 |
45 | elfz2nn0 9128 | . 2 | |
46 | 44, 45 | sylibr 132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wcel 1433 class class class wbr 3785 (class class class)co 5532 cr 6980 cc0 6981 caddc 6984 cle 7154 cmin 7279 cn0 8288 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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