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Mirrors > Home > ILE Home > Th. List > ubmelm1fzo | Unicode version |
Description: The result of subtracting 1 and an integer of a half-open range of nonnegative integers from the upper bound of this range is contained in this range. (Contributed by AV, 23-Mar-2018.) (Revised by AV, 30-Oct-2018.) |
Ref | Expression |
---|---|
ubmelm1fzo | ..^ ..^ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfzo0 10091 | . 2 ..^ | |
2 | nnz 9192 | . . . . . . . . 9 | |
3 | 2 | adantr 274 | . . . . . . . 8 |
4 | nn0z 9193 | . . . . . . . . 9 | |
5 | 4 | adantl 275 | . . . . . . . 8 |
6 | 3, 5 | zsubcld 9297 | . . . . . . 7 |
7 | 6 | ancoms 266 | . . . . . 6 |
8 | peano2zm 9211 | . . . . . 6 | |
9 | 7, 8 | syl 14 | . . . . 5 |
10 | 9 | 3adant3 1002 | . . . 4 |
11 | simp3 984 | . . . . . 6 | |
12 | 4, 2 | anim12i 336 | . . . . . . . 8 |
13 | 12 | 3adant3 1002 | . . . . . . 7 |
14 | znnsub 9224 | . . . . . . 7 | |
15 | 13, 14 | syl 14 | . . . . . 6 |
16 | 11, 15 | mpbid 146 | . . . . 5 |
17 | nnm1ge0 9256 | . . . . 5 | |
18 | 16, 17 | syl 14 | . . . 4 |
19 | elnn0z 9186 | . . . 4 | |
20 | 10, 18, 19 | sylanbrc 414 | . . 3 |
21 | simp2 983 | . . 3 | |
22 | nncn 8847 | . . . . . . 7 | |
23 | 22 | adantl 275 | . . . . . 6 |
24 | nn0cn 9106 | . . . . . . 7 | |
25 | 24 | adantr 274 | . . . . . 6 |
26 | 1cnd 7897 | . . . . . 6 | |
27 | 23, 25, 26 | subsub4d 8222 | . . . . 5 |
28 | nn0p1gt0 9125 | . . . . . . 7 | |
29 | 28 | adantr 274 | . . . . . 6 |
30 | nn0re 9105 | . . . . . . . 8 | |
31 | peano2re 8016 | . . . . . . . 8 | |
32 | 30, 31 | syl 14 | . . . . . . 7 |
33 | nnre 8846 | . . . . . . 7 | |
34 | ltsubpos 8334 | . . . . . . 7 | |
35 | 32, 33, 34 | syl2an 287 | . . . . . 6 |
36 | 29, 35 | mpbid 146 | . . . . 5 |
37 | 27, 36 | eqbrtrd 3989 | . . . 4 |
38 | 37 | 3adant3 1002 | . . 3 |
39 | elfzo0 10091 | . . 3 ..^ | |
40 | 20, 21, 38, 39 | syl3anbrc 1166 | . 2 ..^ |
41 | 1, 40 | sylbi 120 | 1 ..^ ..^ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 963 wcel 2128 class class class wbr 3967 (class class class)co 5827 cc 7733 cr 7734 cc0 7735 c1 7736 caddc 7738 clt 7915 cle 7916 cmin 8051 cn 8839 cn0 9096 cz 9173 ..^cfzo 10051 |
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-csb 3032 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-iun 3853 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-1st 6091 df-2nd 6092 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 df-fzo 10052 |
This theorem is referenced by: (None) |
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