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Mirrors > Home > MPE Home > Th. List > fznn0 | Structured version Visualization version GIF version |
Description: Characterization of a finite set of sequential nonnegative integers. (Contributed by NM, 1-Aug-2005.) |
Ref | Expression |
---|---|
fznn0 | ⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12568 | . . 3 ⊢ 0 ∈ ℤ | |
2 | nn0z 12582 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ) | |
3 | elfz1 13490 | . . 3 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) | |
4 | 1, 2, 3 | sylancr 586 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁))) |
5 | df-3an 1086 | . . 3 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) ↔ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ 𝐾 ≤ 𝑁)) | |
6 | elnn0z 12570 | . . . 4 ⊢ (𝐾 ∈ ℕ0 ↔ (𝐾 ∈ ℤ ∧ 0 ≤ 𝐾)) | |
7 | 6 | anbi1i 623 | . . 3 ⊢ ((𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁) ↔ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾) ∧ 𝐾 ≤ 𝑁)) |
8 | 5, 7 | bitr4i 278 | . 2 ⊢ ((𝐾 ∈ ℤ ∧ 0 ≤ 𝐾 ∧ 𝐾 ≤ 𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁)) |
9 | 4, 8 | bitrdi 287 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐾 ∈ (0...𝑁) ↔ (𝐾 ∈ ℕ0 ∧ 𝐾 ≤ 𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5139 (class class class)co 7402 0cc0 11107 ≤ cle 11248 ℕ0cn0 12471 ℤcz 12557 ...cfz 13485 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-n0 12472 df-z 12558 df-fz 13486 |
This theorem is referenced by: nn0fz0 13600 swrdfv2 14613 efgredlem 19663 coe1mul2lem1 22130 coe1tmmul2 22139 coe1tmmul 22140 coe1mul3 25979 plypf1 26089 dvdsppwf1o 27058 dchrisumlem1 27362 signstfveq0 34107 hbt 42422 |
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