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Mirrors > Home > ILE Home > Th. List > elfzmlbmOLD | GIF 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.) Obsolete version of elfzmlbm 8988 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
elfzmlbmOLD | ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znn0sub 8309 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ≤ 𝐾 ↔ (𝐾 − 𝑀) ∈ ℕ0)) | |
2 | 1 | biimpd 132 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ≤ 𝐾 → (𝐾 − 𝑀) ∈ ℕ0)) |
3 | 2 | 3adant2 923 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ≤ 𝐾 → (𝐾 − 𝑀) ∈ ℕ0)) |
4 | 3 | com12 27 | . . . . 5 ⊢ (𝑀 ≤ 𝐾 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 − 𝑀) ∈ ℕ0)) |
5 | 4 | adantr 261 | . . . 4 ⊢ ((𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 − 𝑀) ∈ ℕ0)) |
6 | 5 | impcom 116 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝐾 − 𝑀) ∈ ℕ0) |
7 | zre 8249 | . . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
8 | zre 8249 | . . . . . . . 8 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ ℝ) | |
9 | zre 8249 | . . . . . . . 8 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
10 | 7, 8, 9 | 3anim123i 1089 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
11 | 10 | 3com23 1110 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ)) |
12 | letr 7101 | . . . . . 6 ⊢ ((𝑀 ∈ ℝ ∧ 𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → 𝑀 ≤ 𝑁)) | |
13 | 11, 12 | syl 14 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → 𝑀 ≤ 𝑁)) |
14 | znn0sub 8309 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ (𝑁 − 𝑀) ∈ ℕ0)) | |
15 | 14 | biimpd 132 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 → (𝑁 − 𝑀) ∈ ℕ0)) |
16 | 15 | 3adant3 924 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ≤ 𝑁 → (𝑁 − 𝑀) ∈ ℕ0)) |
17 | 13, 16 | syld 40 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → (𝑁 − 𝑀) ∈ ℕ0)) |
18 | 17 | imp 115 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝑁 − 𝑀) ∈ ℕ0) |
19 | 8, 9, 7 | 3anim123i 1089 | . . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ)) |
20 | 19 | 3com13 1109 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ)) |
21 | lesub1 7451 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℝ ∧ 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝐾 ≤ 𝑁 ↔ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) | |
22 | 20, 21 | syl 14 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 ≤ 𝑁 ↔ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) |
23 | 22 | biimpd 132 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 ≤ 𝑁 → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) |
24 | 23 | com12 27 | . . . . 5 ⊢ (𝐾 ≤ 𝑁 → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) |
25 | 24 | adantl 262 | . . . 4 ⊢ ((𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) |
26 | 25 | impcom 116 | . . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → (𝐾 − 𝑀) ≤ (𝑁 − 𝑀)) |
27 | 6, 18, 26 | 3jca 1084 | . 2 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁)) → ((𝐾 − 𝑀) ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ0 ∧ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) |
28 | elfz2 8881 | . 2 ⊢ (𝐾 ∈ (𝑀...𝑁) ↔ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ) ∧ (𝑀 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
29 | elfz2nn0 8973 | . 2 ⊢ ((𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀)) ↔ ((𝐾 − 𝑀) ∈ ℕ0 ∧ (𝑁 − 𝑀) ∈ ℕ0 ∧ (𝐾 − 𝑀) ≤ (𝑁 − 𝑀))) | |
30 | 27, 28, 29 | 3imtr4i 190 | 1 ⊢ (𝐾 ∈ (𝑀...𝑁) → (𝐾 − 𝑀) ∈ (0...(𝑁 − 𝑀))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 ℝcr 6888 0cc0 6889 ≤ cle 7061 − cmin 7182 ℕ0cn0 8181 ℤcz 8245 ...cfz 8874 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 ax-pre-ltirr 6996 ax-pre-ltwlin 6997 ax-pre-lttrn 6998 ax-pre-ltadd 7000 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-1r 6817 df-0 6896 df-1 6897 df-r 6899 df-lt 6902 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 df-uz 8474 df-fz 8875 |
This theorem is referenced by: (None) |
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