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Mirrors > Home > ILE Home > Th. List > zpnn0elfzo1 | GIF version |
Description: Membership of an integer increased by a nonnegative integer in a half- open integer range. (Contributed by Alexander van der Vekens, 22-Sep-2018.) |
Ref | Expression |
---|---|
zpnn0elfzo1 | ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zpnn0elfzo 10152 | . 2 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^((𝑍 + 𝑁) + 1))) | |
2 | zcn 9206 | . . . . 5 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℂ) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑍 ∈ ℂ) |
4 | nn0cn 9134 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
5 | 4 | adantl 275 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℂ) |
6 | 1cnd 7925 | . . . 4 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → 1 ∈ ℂ) | |
7 | 3, 5, 6 | addassd 7931 | . . 3 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝑍 + 𝑁) + 1) = (𝑍 + (𝑁 + 1))) |
8 | 7 | oveq2d 5867 | . 2 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍..^((𝑍 + 𝑁) + 1)) = (𝑍..^(𝑍 + (𝑁 + 1)))) |
9 | 1, 8 | eleqtrd 2249 | 1 ⊢ ((𝑍 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → (𝑍 + 𝑁) ∈ (𝑍..^(𝑍 + (𝑁 + 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2141 (class class class)co 5851 ℂcc 7761 1c1 7764 + caddc 7766 ℕ0cn0 9124 ℤcz 9201 ..^cfzo 10087 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-inn 8868 df-n0 9125 df-z 9202 df-uz 9477 df-fz 9955 df-fzo 10088 |
This theorem is referenced by: (None) |
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