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Mirrors > Home > MPE Home > Th. List > hashfz0 | Structured version Visualization version GIF version |
Description: Value of the numeric cardinality of a nonempty range of nonnegative integers. (Contributed by Alexander van der Vekens, 21-Jul-2018.) |
Ref | Expression |
---|---|
hashfz0 | ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0uz 12568 | . . 3 ⊢ (𝐵 ∈ ℕ0 ↔ 𝐵 ∈ (ℤ≥‘0)) | |
2 | hashfz 14086 | . . 3 ⊢ (𝐵 ∈ (ℤ≥‘0) → (♯‘(0...𝐵)) = ((𝐵 − 0) + 1)) | |
3 | 1, 2 | sylbi 216 | . 2 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = ((𝐵 − 0) + 1)) |
4 | nn0cn 12189 | . . . 4 ⊢ (𝐵 ∈ ℕ0 → 𝐵 ∈ ℂ) | |
5 | 4 | subid1d 11267 | . . 3 ⊢ (𝐵 ∈ ℕ0 → (𝐵 − 0) = 𝐵) |
6 | 5 | oveq1d 7275 | . 2 ⊢ (𝐵 ∈ ℕ0 → ((𝐵 − 0) + 1) = (𝐵 + 1)) |
7 | 3, 6 | eqtrd 2777 | 1 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ‘cfv 6423 (class class class)co 7260 0cc0 10818 1c1 10819 + caddc 10821 − cmin 11151 ℕ0cn0 12179 ℤ≥cuz 12527 ...cfz 13184 ♯chash 13988 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 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 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 ax-cnex 10874 ax-resscn 10875 ax-1cn 10876 ax-icn 10877 ax-addcl 10878 ax-addrcl 10879 ax-mulcl 10880 ax-mulrcl 10881 ax-mulcom 10882 ax-addass 10883 ax-mulass 10884 ax-distr 10885 ax-i2m1 10886 ax-1ne0 10887 ax-1rid 10888 ax-rnegex 10889 ax-rrecex 10890 ax-cnre 10891 ax-pre-lttri 10892 ax-pre-lttrn 10893 ax-pre-ltadd 10894 ax-pre-mulgt0 10895 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3429 df-sbc 3717 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6259 df-on 6260 df-lim 6261 df-suc 6262 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-f1 6428 df-fo 6429 df-f1o 6430 df-fv 6431 df-riota 7217 df-ov 7263 df-oprab 7264 df-mpo 7265 df-om 7693 df-1st 7809 df-2nd 7810 df-frecs 8073 df-wrecs 8104 df-recs 8178 df-rdg 8217 df-1o 8272 df-er 8461 df-en 8697 df-dom 8698 df-sdom 8699 df-fin 8700 df-card 9644 df-pnf 10958 df-mnf 10959 df-xr 10960 df-ltxr 10961 df-le 10962 df-sub 11153 df-neg 11154 df-nn 11920 df-n0 12180 df-z 12266 df-uz 12528 df-fz 13185 df-hash 13989 |
This theorem is referenced by: fnfz0hash 14102 pthhashvtx 33031 poimirlem25 35771 poimirlem26 35772 lcmineqlem17 40023 etransclem23 43730 lighneallem3 44989 lighneallem4 44992 aacllem 46435 |
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