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Mirrors > Home > ILE Home > Th. List > elfz0addOLD | GIF version |
Description: An element of a finite set of sequential nonnegative integers is an element of an extended finite set of sequential nonnegative integers. (Contributed by Alexander van der Vekens, 28-Mar-2018.) Obsolete version of elfz0add 8979 as of 25-Mar-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
elfz0addOLD | ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0z 8265 | . . . . 5 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℤ) | |
2 | 1 | adantr 261 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → 𝐴 ∈ ℤ) |
3 | nn0addcl 8217 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℕ0) | |
4 | 3 | nn0zd 8358 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ ℤ) |
5 | nn0ge0 8207 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 0 ≤ 𝐵) | |
6 | 5 | adantl 262 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → 0 ≤ 𝐵) |
7 | nn0re 8190 | . . . . . 6 ⊢ (𝐴 ∈ ℕ0 → 𝐴 ∈ ℝ) | |
8 | nn0re 8190 | . . . . . 6 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℝ) | |
9 | addge01 7467 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) | |
10 | 7, 8, 9 | syl2an 273 | . . . . 5 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (0 ≤ 𝐵 ↔ 𝐴 ≤ (𝐴 + 𝐵))) |
11 | 6, 10 | mpbid 135 | . . . 4 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → 𝐴 ≤ (𝐴 + 𝐵)) |
12 | eluz2 8479 | . . . 4 ⊢ ((𝐴 + 𝐵) ∈ (ℤ≥‘𝐴) ↔ (𝐴 ∈ ℤ ∧ (𝐴 + 𝐵) ∈ ℤ ∧ 𝐴 ≤ (𝐴 + 𝐵))) | |
13 | 2, 4, 11, 12 | syl3anbrc 1088 | . . 3 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝐴 + 𝐵) ∈ (ℤ≥‘𝐴)) |
14 | fzss2 8927 | . . 3 ⊢ ((𝐴 + 𝐵) ∈ (ℤ≥‘𝐴) → (0...𝐴) ⊆ (0...(𝐴 + 𝐵))) | |
15 | 13, 14 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (0...𝐴) ⊆ (0...(𝐴 + 𝐵))) |
16 | 15 | sseld 2944 | 1 ⊢ ((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) → (𝑁 ∈ (0...𝐴) → 𝑁 ∈ (0...(𝐴 + 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ⊆ wss 2917 class class class wbr 3764 ‘cfv 4902 (class class class)co 5512 ℝcr 6888 0cc0 6889 + caddc 6892 ≤ cle 7061 ℕ0cn0 8181 ℤcz 8245 ℤ≥cuz 8473 ...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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