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Mirrors > Home > MPE Home > Th. List > nn0split | Structured version Visualization version GIF version |
Description: Express the set of nonnegative integers as the disjoint (see nn0disj 13623) union of the first 𝑁 + 1 values and the rest. (Contributed by AV, 8-Nov-2019.) |
Ref | Expression |
---|---|
nn0split | ⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0uz 12868 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑁 ∈ ℕ0 → ℕ0 = (ℤ≥‘0)) |
3 | peano2nn0 12516 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
4 | 3, 1 | eleqtrdi 2837 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ≥‘0)) |
5 | uzsplit 13579 | . . 3 ⊢ ((𝑁 + 1) ∈ (ℤ≥‘0) → (ℤ≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ≥‘0) = ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1)))) |
7 | nn0cn 12486 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
8 | pncan1 11642 | . . . . 5 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 + 1) − 1) = 𝑁) |
10 | 9 | oveq2d 7421 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0...((𝑁 + 1) − 1)) = (0...𝑁)) |
11 | 10 | uneq1d 4157 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((0...((𝑁 + 1) − 1)) ∪ (ℤ≥‘(𝑁 + 1))) = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
12 | 2, 6, 11 | 3eqtrd 2770 | 1 ⊢ (𝑁 ∈ ℕ0 → ℕ0 = ((0...𝑁) ∪ (ℤ≥‘(𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ‘cfv 6537 (class class class)co 7405 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11448 ℕ0cn0 12476 ℤ≥cuz 12826 ...cfz 13490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 |
This theorem is referenced by: chfacfscmulgsum 22717 chfacfpmmulgsum 22721 bccbc 43680 |
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