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Mirrors > Home > MPE Home > Th. List > hashfzo0 | Structured version Visualization version GIF version |
Description: Cardinality of a half-open set of integers based at zero. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
hashfzo0 | ⊢ (𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashfzo 13786 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘0) → (♯‘(0..^𝐵)) = (𝐵 − 0)) | |
2 | nn0uz 12268 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
3 | 1, 2 | eleq2s 2908 | . 2 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = (𝐵 − 0)) |
4 | nn0cn 11895 | . . 3 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ) | |
5 | 4 | subid1d 10975 | . 2 ⊢ (𝐵 ∈ ℕ0 → (𝐵 − 0) = 𝐵) |
6 | 3, 5 | eqtrd 2833 | 1 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0..^𝐵)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 0cc0 10526 − cmin 10859 ℕ0cn0 11885 ℤ≥cuz 12231 ..^cfzo 13028 ♯chash 13686 |
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-int 4839 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-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 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 df-hash 13687 |
This theorem is referenced by: ffzo0hash 13803 hashwrdn 13890 eqwrd 13900 wrdred1hash 13904 ccatlen 13918 ccatlenOLD 13919 ccatalpha 13938 swrdlen 14000 swrdwrdsymb 14015 pfxlen 14036 revlen 14115 repswlen 14129 ofccat 14320 crth 16105 phisum 16117 cshwshashnsame 16429 pmtrdifwrdellem2 18602 odhash2 18692 ablfaclem3 19202 znhash 20250 cycpmconjslem2 30847 subiwrdlen 31754 ccatmulgnn0dir 31922 ofcccat 31923 signstlen 31947 signsvtn0 31950 signstres 31955 signshlen 31970 reprlt 32000 reprgt 32002 breprexpnat 32015 circlemethnat 32022 circlevma 32023 hgt750lema 32038 lpadlem2 32061 frlmvscadiccat 39440 fltnltalem 39618 amgm2d 40904 amgm3d 40905 amgm4d 40906 fourierdlem73 42821 |
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