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Mirrors > Home > MPE Home > Th. List > fz0sn0fz1 | Structured version Visualization version GIF version |
Description: A finite set of sequential nonnegative integers is the union of the singleton containing 0 and a finite set of sequential positive integers. (Contributed by AV, 20-Mar-2021.) |
Ref | Expression |
---|---|
fz0sn0fz1 | ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elfz 12818 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
2 | fzsplit 12747 | . . . 4 ⊢ (0 ∈ (0...𝑁) → (0...𝑁) = ((0...0) ∪ ((0 + 1)...𝑁))) | |
3 | 0p1e1 11567 | . . . . . 6 ⊢ (0 + 1) = 1 | |
4 | 3 | oveq1i 6984 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
5 | 4 | uneq2i 4018 | . . . 4 ⊢ ((0...0) ∪ ((0 + 1)...𝑁)) = ((0...0) ∪ (1...𝑁)) |
6 | 2, 5 | syl6eq 2823 | . . 3 ⊢ (0 ∈ (0...𝑁) → (0...𝑁) = ((0...0) ∪ (1...𝑁))) |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ((0...0) ∪ (1...𝑁))) |
8 | 0z 11802 | . . . 4 ⊢ 0 ∈ ℤ | |
9 | fzsn 12763 | . . . 4 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0...0) = {0}) |
11 | 10 | uneq1d 4020 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((0...0) ∪ (1...𝑁)) = ({0} ∪ (1...𝑁))) |
12 | 7, 11 | eqtrd 2807 | 1 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ∪ cun 3820 {csn 4435 (class class class)co 6974 0cc0 10333 1c1 10334 + caddc 10336 ℕ0cn0 11705 ℤcz 11791 ...cfz 12706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-fz 12707 |
This theorem is referenced by: eucrct2eupthOLD 27791 eucrct2eupth 27792 |
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