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Mirrors > Home > ILE Home > Th. List > elfzmlbm | 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.) (Proof shortened by OpenAI, 25-Mar-2020.) |
Ref | Expression |
---|---|
elfzmlbm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzuz 9931 | . . 3 | |
2 | uznn0sub 9476 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | elfzuz2 9938 | . . 3 | |
5 | uznn0sub 9476 | . . 3 | |
6 | 4, 5 | syl 14 | . 2 |
7 | elfzelz 9935 | . . . 4 | |
8 | 7 | zred 9292 | . . 3 |
9 | elfzel2 9933 | . . . 4 | |
10 | 9 | zred 9292 | . . 3 |
11 | elfzel1 9934 | . . . 4 | |
12 | 11 | zred 9292 | . . 3 |
13 | elfzle2 9937 | . . 3 | |
14 | 8, 10, 12, 13 | lesub1dd 8441 | . 2 |
15 | elfz2nn0 10021 | . 2 | |
16 | 3, 6, 14, 15 | syl3anbrc 1166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2128 class class class wbr 3967 cfv 5173 (class class class)co 5827 cc0 7735 cle 7916 cmin 8051 cn0 9096 cuz 9445 cfz 9919 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-addcom 7835 ax-addass 7837 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-0id 7843 ax-rnegex 7844 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-ltadd 7851 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-inn 8840 df-n0 9097 df-z 9174 df-uz 9446 df-fz 9920 |
This theorem is referenced by: bcm1k 10646 |
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