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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 13539 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁)) | |
2 | fzsplit 13468 | . . . 4 ⊢ (0 ∈ (0...𝑁) → (0...𝑁) = ((0...0) ∪ ((0 + 1)...𝑁))) | |
3 | 0p1e1 12276 | . . . . . 6 ⊢ (0 + 1) = 1 | |
4 | 3 | oveq1i 7368 | . . . . 5 ⊢ ((0 + 1)...𝑁) = (1...𝑁) |
5 | 4 | uneq2i 4121 | . . . 4 ⊢ ((0...0) ∪ ((0 + 1)...𝑁)) = ((0...0) ∪ (1...𝑁)) |
6 | 2, 5 | eqtrdi 2793 | . . 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 12511 | . . . 4 ⊢ 0 ∈ ℤ | |
9 | fzsn 13484 | . . . 4 ⊢ (0 ∈ ℤ → (0...0) = {0}) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (0...0) = {0}) |
11 | 10 | uneq1d 4123 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((0...0) ∪ (1...𝑁)) = ({0} ∪ (1...𝑁))) |
12 | 7, 11 | eqtrd 2777 | 1 ⊢ (𝑁 ∈ ℕ0 → (0...𝑁) = ({0} ∪ (1...𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cun 3909 {csn 4587 (class class class)co 7358 0cc0 11052 1c1 11053 + caddc 11055 ℕ0cn0 12414 ℤcz 12500 ...cfz 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-n0 12415 df-z 12501 df-uz 12765 df-fz 13426 |
This theorem is referenced by: eucrct2eupth 29192 f1resfz0f1d 33707 |
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