![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fzo0end | Structured version Visualization version GIF version |
Description: The endpoint of a zero-based half-open range. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Revised by Mario Carneiro, 29-Sep-2015.) |
Ref | Expression |
---|---|
fzo0end | ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ (0..^𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lbfzo0 13072 | . 2 ⊢ (0 ∈ (0..^𝐵) ↔ 𝐵 ∈ ℕ) | |
2 | fzoend 13123 | . 2 ⊢ (0 ∈ (0..^𝐵) → (𝐵 − 1) ∈ (0..^𝐵)) | |
3 | 1, 2 | sylbir 238 | 1 ⊢ (𝐵 ∈ ℕ → (𝐵 − 1) ∈ (0..^𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 (class class class)co 7135 0cc0 10526 1c1 10527 − cmin 10859 ℕcn 11625 ..^cfzo 13028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 |
This theorem is referenced by: lswcl 13911 ccatval1lsw 13929 swrdlsw 14020 pfxfvlsw 14048 pfxsuff1eqwrdeq 14052 wrdind 14075 wrd2ind 14076 repswlsw 14135 cshwidxn 14162 lswco 14192 swrd2lsw 14305 efgsf 18847 efgsrel 18852 efgsp1 18855 efgredlemf 18859 efgredlemd 18862 efgredlemc 18863 efgredlem 18865 taylthlem1 24968 wlkdlem2 27473 pthdlem2lem 27556 clwwlkel 27831 clwwlkf 27832 clwwlkwwlksb 27839 eucrct2eupth1 28029 2clwwlk2clwwlklem 28131 cycpmco2lem5 30822 fiblem 31766 signstfvn 31949 signsvtn0 31950 signstfvneq0 31952 signstfveq0 31957 signsvfn 31962 signsvtp 31963 signsvtn 31964 signsvfpn 31965 signsvfnn 31966 signlem0 31967 |
Copyright terms: Public domain | W3C validator |