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Mirrors > Home > ILE Home > Th. List > nnsplit | GIF version |
Description: Express the set of positive integers as the disjoint union of the first 𝑁 values and the rest. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
Ref | Expression |
---|---|
nnsplit | ⊢ (𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9592 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1 | a1i 9 | . 2 ⊢ (𝑁 ∈ ℕ → ℕ = (ℤ≥‘1)) |
3 | peano2nn 8960 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
4 | 3, 1 | eleqtrdi 2282 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ (ℤ≥‘1)) |
5 | uzsplit 10121 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘1) → (ℤ≥‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → (ℤ≥‘1) = ((1...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))) |
7 | nncn 8956 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
8 | 1cnd 8002 | . . . . 5 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) | |
9 | 7, 8 | pncand 8298 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((𝑁 + 1) − 1) = 𝑁) |
10 | 9 | oveq2d 5911 | . . 3 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 + 1) − 1)) = (1...𝑁)) |
11 | 10 | uneq1d 3303 | . 2 ⊢ (𝑁 ∈ ℕ → ((1...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))) = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
12 | 2, 6, 11 | 3eqtrd 2226 | 1 ⊢ (𝑁 ∈ ℕ → ℕ = ((1...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ∪ cun 3142 ‘cfv 5235 (class class class)co 5895 1c1 7841 + caddc 7843 − cmin 8157 ℕcn 8948 ℤ≥cuz 9557 ...cfz 10037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-addass 7942 ax-distr 7944 ax-i2m1 7945 ax-0lt1 7946 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951 ax-pre-ltirr 7952 ax-pre-ltwlin 7953 ax-pre-lttrn 7954 ax-pre-ltadd 7956 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-pnf 8023 df-mnf 8024 df-xr 8025 df-ltxr 8026 df-le 8027 df-sub 8159 df-neg 8160 df-inn 8949 df-n0 9206 df-z 9283 df-uz 9558 df-fz 10038 |
This theorem is referenced by: summodclem3 11419 |
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