Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 13174 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
2 | fzsplit 13103 | . . . 4 ⊢ (0 ∈ (0...𝑁) → (0...𝑁) = ((0...0) ∪ ((0 + 1)...𝑁))) | |
3 | 0p1e1 11917 | . . . . . 6 ⊢ (0 + 1) = 1 | |
4 | 3 | oveq1i 7201 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
5 | 4 | uneq2i 4060 | . . . 4 ⊢ ((0...0) ∪ ((0 + 1)...𝑁)) = ((0...0) ∪ (1...𝑁)) |
6 | 2, 5 | eqtrdi 2787 | . . 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 12152 | . . . 4 ⊢ 0 ∈ ℤ | |
9 | fzsn 13119 | . . . 4 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0...0) = {0}) |
11 | 10 | uneq1d 4062 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((0...0) ∪ (1...𝑁)) = ({0} ∪ (1...𝑁))) |
12 | 7, 11 | eqtrd 2771 | 1 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ∪ cun 3851 {csn 4527 (class class class)co 7191 0cc0 10694 1c1 10695 + caddc 10697 ℕ0cn0 12055 ℤcz 12141 ...cfz 13060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 |
This theorem is referenced by: eucrct2eupth 28282 f1resfz0f1d 32739 |
Copyright terms: Public domain | W3C validator |